Literature Survey on Non-Associative Rings and Developments
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paper_title: Historical notes on loop theory

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This paper deals with the origins and early history of loop theory, summarizing the period from the 1920s through the 1960s.
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paper_title: Congruences on left almost rings

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The aim of this paper is to characterize left almost rings by congrunces. We show that each homomophism of left amost rings defines a congrucne relation on left almost rings. We then discuss quotient left almiost rings. At the end we prove analogues of the ismorphism theorem for left almost rings.
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paper_title: The Wedderburn principal theorem for alternative algebras

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for all a, x in 21. I t is clear that associative algebras are alternative. The most famous examples of alternative algebras which are not associative are the so-called Cayley-Dickson algebras of order 8 over $. Let S be an algebra of order 2 over % which is either a separable quadratic field over 5 or the direct sum 5 ©3There is one automorphism z—>z of S (over %) which is not the identity automorphism. The associative algebra O = 3~\~S with elements
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paper_title: Some results in the theory of linear non-associative algebras

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A waterproof pressure sensitive adhesive laminate is provided in which a flexible plastics backing sheet is coated with a bituminous adhesive composition containing a minor proportion of rubber or thermoplastic polymer. The backing sheet is reinforced with a mesh or a woven or non-woven fabric which is embedded in the sheet and provides substantial resistance to stretching.
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paper_title: Inner derivations of non-associative algebras

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A compression device for exerting pressure on an arm, shoulder, and/or trunk of a patient in need thereof (for example, a patient with hyperalgia or recovering from surgery in which the lymphatic system is affected), including an arm compression hose, a shoulder part for exerting pressure on the shoulder and trunk area, and a band-shaped fastening means for positioning the shoulder part and exerting pressure on the shoulder part. The arm compression hose exerts a pressure that decreases from a maximum pressure at the wrist or hand to a minimum pressure near the shoulder end of the arm, where the minimum pressure is approximately 70% of the maximum pressure. One or more lining pockets can be constructed on the inner lining of the compression device, where each lining pocket can hold one or more compression pads to increase tissue pressure in one or more body areas in need thereof. The compression pads each can have a shape that approximately conforms to the shape of the body part to which it is applied. The shoulder part can also have a shape that approximately conforms to the contour of the shoulder/trunk area to which it is applied. In addition, compression pants can be prepared with lining pockets for receiving compression pads. In one embodiment, compression pants include one or more donut-shaped pads or equivalents thereof that are placed in one or more lining pockets, each of which surrounds one or more osteoma openings.
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paper_title: Group Theory in Physics: An Introduction

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The Basic Framework. The Structure of Groups. Lie Groups. Representation of Groups--Principal Ideas. Representation of Groups--Developments. Group Theory in Quantum Mechanical Calculations. Crystallographic Space Groups. The Role of Lie Algebras. The Relationships Between Lie Groups and Lie Algebras Explored. The Three-Dimensional Rotation Groups. The Structure of Semi-Simple Lie Algebras. Representations of Semi-Simple Lie Algebras. Symmetry Schemes for the Elementary Particles. Appendices. References. Subject Index.
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paper_title: Theorie der alternativen ringe

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A new catalyst component and its use with an organoaluminum compound, which component is a brown solid of high surface area and large pore volume comprising beta titanium trichloride and a small amount of an organic electron pair donor compound. This solid when used in conjunction with an organoaluminum compound to polymerize alpha-olefins produces product polymer at substantially increased rates and yields compared to present commercial, purple titanium trichloride while coproducing reduced amounts of low-molecular-weight and, particularly, amorphous polymer. Combinations of this new catalyst component and an organoaluminum compound can be further improved in their catalytic properties by addition of small amounts of modifiers, alone and in combination. Such combinations with or without modifiers show good sensitivity to hydrogen used as a molecular weight controlling agent. The combinations are useful for slurry, bulk and vapor phase polymerization of alpha-olefins such as propylene.
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paper_title: On Left Primary and Weakly Left Primary Ideals in LA- Rings

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In this paper, we study left ideals, left primary and weakly left primary ideals in LA-rings. Some characterizations of left primary and weakly left primary ideals are obtained. Moreover, we investigate relationships left primary and weakly left primary ideals in LA-rings. Finally, we obtain necessary and sufficient conditions of a weakly left primary ideal to be a left primary ideals in LA- rings.
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paper_title: Some results in the theory of quasigroups

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Introduction. The concept of isotopy, recently introduced(') by A. A. Albert in connection with the theory of linear non-associative algebras, appears to have its value in the theory of quasigroups. Conversely, the author has been able to use quasigroups(2) in the study of linear non-associative algebras. The present paper is primarily intended as an illustration of the usefulness of isotopy in quasigroup-theory and as groundwork for a later paper on algebras, but is bounded by neither of these aspects. The first two sections are devoted to the basic definitions of quasigroup and isotopy, along with some elementary remarks and two fundamental theorems due to Albert. Then there is initiated a study of special types of quasigroup, beginning with quasigroups with the inverse property (I. P. quasigroups). A system Q of elements a, b, * * * is called an I. P. quasigroup if it possesses a single-valued binary operation ab and there exist two one-to-one reversible mappings L and R of Q on itself such that
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paper_title: Introduction To Lie Algebras And Representation Theory

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Thank you very much for downloading introduction to lie algebras and representation theory. As you may know, people have search numerous times for their favorite books like this introduction to lie algebras and representation theory, but end up in harmful downloads. Rather than enjoying a good book with a cup of tea in the afternoon, instead they cope with some harmful virus inside their desktop computer.
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paper_title: Jordan structures in mathematics and physics

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The aim of this paper is to offer an overview of the most important applications of Jordan structures inside mathematics and also to physics, up-dated references being included. For a more detailed treatment of this topic see - especially - the recent book Iordanescu [364w], where sugestions for further developments are given through many open problems, comments and remarks pointed out throughout the text. ::: Nowadays, mathematics becomes more and more nonassociative and my prediction is that in few years nonassociativity will govern mathematics and applied sciences. ::: Keywords: Jordan algebra, Jordan triple system, Jordan pair, JB-, JB*-, JBW-, JBW*-, JH*-algebra, Ricatti equation, Riemann space, symmetric space, R-space, octonion plane, projective plane, Barbilian space, Tzitzeica equation, quantum group, B\"acklund-Darboux transformation, Hopf algebra, Yang-Baxter equation, KP equation, Sato Grassmann manifold, genetic algebra, random quadratic form.
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paper_title: Quaternions and Cayley Numbers

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Oil from subsurface tar sand having an injection means in fluid communication with a production means is recovered by injecting a water-external micellar dispersion at a temperature above 100 DEG F., into the tar sands, displacing it toward the production means and recovering the oil through the production means. The micellar dispersion can be preceded by a slug of hot water which can optionally have a pH greater than about 7. Also, the micellar dispersion can have a pH of about 7-14 and preferably a temperature greater than about 150 DEG F. The micellar dispersion contains hydrocarbon, surfactant, aqueous medium, and optionally cosurfactant and/or electrolyte.
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paper_title: Division Algebras:: Octonions Quaternions Complex Numbers and the Algebraic Design of Physics

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I. Underpinnings. II. Division Algebra Alone. III. Tensor Algebras. IV. Connecting to Physics. V. Spontaneous Symmetry Breaking. VI. 10 Dimensions. VII. Doorways. VIII. Corridors. Appendices. Bibliography. Index.
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paper_title: On Quaternions and Octonions

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This book investigates the geometry of quaternion and octonion algebras. Following a comprehensive historical introduction, the book illuminates the special properties of 3- and 4-dimensional Euclidean spaces using quaternions, leading to enumerations of the corresponding finite groups of symmetries. The second half of the book discusses the less familiar octonion algebra, concentrating on its remarkable "triality symmetry" after an appropriate study of Moufang loops. The authors also describe the arithmetics of the quaternions and octonions. The book concludes with a new theory of octonion factorization. Topics covered include the geometry of complex numbers, quaternions and 3-dimensional groups, quaternions and 4-dimensional groups, Hurwitz integral quaternions, composition algebras, Moufang loops, octonions and 8-dimensional geometry, integral octonions, and the octonion projective plane.
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paper_title: COMPOSITION ALGEBRAS OVER ALGEBRAIC CURVES OF GENUS ZERO

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We rephrase the classical theory of composition algebras over fields, particularly the Cayley-Dickson Doubling Process and Zorn's Vector Matrices, in the setting of locally ringed spaces. Fixing an arbitrary base field, we use these constructions to classify composition algebras over (complete smooth) curves of genus zero. Applications are given to composition algebras over function fields of genus zero and polynomial rings.
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paper_title: Introduction to Octonion and Other Non-Associative Algebras in Physics

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1. Introduction 2. Non-associative algebras 3. Hurwitz theorems and octonions 4. Para-Hurwitz and pseudo-octonion algebras 5. Real division algebras and Clifford algebra 6. Clebsch-Gordon algebras 7. Algebra of physical observables 8. Triple products and ternary systems 9. Non-associative gauge theory 10. Concluding remarks.
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paper_title: Lie rings with almost regular automorphisms

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Let L be a Lie ring or a Lie algebra of arbitrary, not necessarily finite, dimension. Let φ be an automorphism of L and let CL(φ) = {a ∈ L | φ(a) = a} denote the fixed-point subring. The automorphism φ is called regular if CL(φ)= 0, that is, φ has no non-trivial fixed points. By Kreknin’s theorem [20] if a Lie ring L admits a regular automorphism φ of finite order k, that is, such that φ = 1 and CL(φ)= 0, then L is soluble of derived length bounded by a function of k, actually, by 2 − 2. (Earlier Borel and Mostow [3] proved the solubility in the finite-dimensional case, without a bound for the derived length.) In the present paper we prove that if a Lie ring admits an automorphism of prime-power order that is “almost regular,” then L is “almost soluble.”
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paper_title: Lie Structure in Simple Gamma Rings

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The object of this paper is to study Lie structure in simple gamma rings. We obtain some structural results of simple gamma rings with Lie ideals.
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paper_title: ON THE RELATIONSHIP BETWEEN THE CLASS OF A LIE ALGEBRA AND THE CLASSES OF ITS SUBALGEBRAS

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A classical nilpotency result considers finite p-groups whose proper subgroups all have class bounded by a fixed number n. We consider the analogous property in nilpotent Lie algebras. In particular, we investigate whether this condition puts a bound on the class of the Lie algebra. Some p-group results and proofs carry over directly to the Lie algebra case, some carry over with modified proofs and some fail. For the final of these cases, a certain metabelian Lie algebra is constructed to show a case when the p-groups and Lie algebra cases differ.
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paper_title: Lie and Jordan Structure in Simple Gamma Rings

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In this paper, we study Lie and Jordan structures in simple Γ-rings of characteristic not equal to two. Some properties of these Γ-rings are developed.
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paper_title: On the Lie and Jordan Rings of a Simple Associative Ring

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Given any associative ring A we can form, using its operations and its elements, two new rings. These use the elements of A and the addition as defined in A, but new multiplications are introduced to render them rings, albeit not necessarily associative rings. The first of these, the Lie ring AL of A uses a multiplication defined by [a, b] =ab ba for any a, b e A where ab is the ordinary associative product of elements in A. The second of these, the Jordan ring of A, A', has its multiplication defined by ao bab + ba for any pair of elements a, b in A. Being defined in a manner so decidedly dependent on the associative product of A, it is natural to expect that an intimate relationship should exist between the structure of these two new rings and that of A. In this paper we study one phase of this relationship, namely the connection between the ideal structure of A as an associative ring with the ideal structure of AL and A' as Lie and Jordan rings respectively. To be more specific, we investigate how simplicity of A as an associative ring reflects into analogous properties of AL and A,. When we say that U is an ideal of Al, or, equivalently, when we say that U is a Jordan ideal of A, we mean that U is an additive subgroup of A and that for any xeU and any yeA, xoy=xy+yx is an element of U. We similarly define Lie ideals of A and ideals of AL. Although the main results of this paper deal with the case in which A is a simple ring, many of the other results do not require the assumption of simplicity in order to remain valid; so, unless otherwise stated, we make no assumption of simplicity for A.
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paper_title: A Nilpotent Ideal in the Lie Rings with Automorphism of Prime Order

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We improve the conclusion in Khukhro's theorem stating that a Lie ring (algebra) L admitting an automorphism of prime order p with finitely many m fixed points (with finite-dimensional fixed-point subalgebra of dimension m) has a subring (subalgebra) H of nilpotency class bounded by a function of p such that the index of the additive subgroup |L: H| (the codimension of H) is bounded by a function of m and p. We prove that there exists an ideal, rather than merely a subring (subalgebra), of nilpotency class bounded in terms of p and of index (codimension) bounded in terms of m and p. The proof is based on the method of generalized, or graded, centralizers which was originally suggested in [E. I. Khukhro, Math. USSR Sbornik 71 (1992) 51–63]. An important precursor is a joint theorem of the author and E. I. Khukhro on almost solubility of Lie rings (algebras) with almost regular automorphisms of finite order.
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paper_title: A non-nilpotent Lie ring satisfying the Engel condition and a non-nilpotent Engel group

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Let L be a Lie ring and denote the product of x and y in L by [ x , y ]. The ring L is said to satisfy the Engel condition (cf. (1)), if for every pair of elements x, y e L there is an integer k = k ( x , y )such that If k ( x , y ) can be taken equal to a fixed integer n for all x , y e L then L is said to satisfy the n-th Engel condition .
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paper_title: On the Representation of Lie Rings in Associative Rings

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V.M. Kurochkin [1] has formulated the following theorem: Every Σ-operator Lie ring L has a faithful representation in an associative Σ-operator ring A, where Σ is an arbitrary domain of operators for the ring L. In a subsequent note [2], V.M. Kurochkin pointed out the insufficient rigor of the proof he proposed for this theorem.
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paper_title: Topics In Ring Theory

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In this chapter we shall make a study of rings satisfying certain ascending chain conditions. In the non-commutative case-and this is really the only case with which we shall be concerned- the decisive and incisive results are three theorems due to Goldie. The main part of the chapter will be taken up with a presentation of these.
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paper_title: Isomorphisms of the Unitriangular Groups and Associated Lie Rings for the Exceptional Dimensions

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Isomorphisms between finitary unitriangular groups and those of associated Lie rings are studied. In this paper we investigate exceptional cases.
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paper_title: A Survey of Lie Groups and Lie Algebras with Applications and Computational Methods

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Introduces the concepts and methods of the Lie theory in a form accesible to the non-specialist by keeping the mathematical prerequisites to a minimum. The book is directed towards the reader seeking a broad view of the subject rather than elaborate information about technical details.
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paper_title: Almost solubility of Lie algebras with almost regular automorphisms

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The well-known theorem of Borel–Mostow–Kreknin on solubility of Lie algebras with regular automorphisms is generalized to the case of almost regular automorphisms. It is proved that if a Lie algebra L admits an automorphism ϕ of finite order n with finite-dimensional fixed-point subalgebra of dimension dimCL(ϕ)=m, then L has a soluble ideal of derived length bounded by a function of n whose codimension is bounded by a function of m and n (Theorem 1). A virtually equivalent formulation is in terms of a (Z/nZ)-graded Lie algebra L whose zero component L0 has finite dimension m. The functions of n and of m and n in Theorem 1 can be given explicit upper estimates. The proof is of combinatorial nature and uses the criterion for solubility of Lie rings with an automorphism obtained in [E.I. Khukhro, Siberian Math. J. 42 (2001) 996–1000]. The method of generalized, or graded, centralizers is developed, which was originally created in [E.I. Khukhro, Math. USSR Sbornik 71 (1992) 51–63] for almost regular automorphisms of prime order. As a corollary we prove a result analogous to Theorem 1 on locally nilpotent torsion-free groups admitting an automorphism of finite order with the fixed points subgroup of finite rank (Theorem 3). We also prove an analogous result for Lie rings with an automorphism of finite order having finitely many fixed points (Theorem 2).
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paper_title: Finite soluble groups containing an element of prime order whose centralizer is small

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In [2] we proved that ifG is a finite group containing an involution whose centralizer has order bounded by some numberm, thenG contains a nilpotent subgroup of class at most two and index bounded in terms ofm. One of the steps in the proof of that result was to show that ifG is soluble, then ¦G/F(G) ¦ is bounded by a function ofm, where F (G) is the Fitting subgroup ofG. We now show that, in this part of the argument, the involution can be replaced by an arbitrary element of prime order.
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paper_title: New Lie products for groups and their automorphisms

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We generalize the common notion of descending and ascending central series. The descending approach determines a naturally graded Lie ring and the ascending version determines a graded module for this ring. We also link derivations of these rings to the automorphisms of a group. This uncovers new structure in 4/5 of the approximately 11.8 million groups of size at most 1000 and beyond that point pertains to at least a positive logarithmic proportion of all finite groups.
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paper_title: Frobenius groups of automorphisms and their fixed points

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Suppose that a finite group $G$ admits a Frobenius group of automorphisms $FH$ with kernel $F$ and complement $H$ such that the fixed-point subgroup of $F$ is trivial: $C_G(F)=1$. In this situation various properties of $G$ are shown to be close to the corresponding properties of $C_G(H)$. By using Clifford's theorem it is proved that the order $|G|$ is bounded in terms of $|H|$ and $|C_G(H)|$, the rank of $G$ is bounded in terms of $|H|$ and the rank of $C_G(H)$, and that $G$ is nilpotent if $C_G(H)$ is nilpotent. Lie ring methods are used for bounding the exponent and the nilpotency class of $G$ in the case of metacyclic $FH$. The exponent of $G$ is bounded in terms of $|FH|$ and the exponent of $C_G(H)$ by using Lazard's Lie algebra associated with the Jennings--Zassenhaus filtration and its connection with powerful subgroups. The nilpotency class of $G$ is bounded in terms of $|H|$ and the nilpotency class of $C_G(H)$ by considering Lie rings with a finite cyclic grading satisfying a certain `selective nilpotency' condition. The latter technique also yields similar results bounding the nilpotency class of Lie rings and algebras with a metacyclic Frobenius group of automorphisms, with corollaries for connected Lie groups and torsion-free locally nilpotent groups with such groups of automorphisms. Examples show that such nilpotency results are no longer true for non-metacyclic Frobenius groups of automorphisms.
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paper_title: Lie Rings of Derivations of Associative Rings

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Let $R$ be an associative ring with centre $Z$. The aim of this paper is to study how the ideal structure of the Lie ring of derivations of $R$, denoted $D(R)$, is determined by the ideal structure of $R$. If $R$ is a simple (respectively semisimple) finite-dimensional $Z$-algebra and δ$(z)$ = 0 for all δ ∈ $D(R)$, then every derivation of $R$ is inner and $D(R)$ is known to be a simple (respectively semisimple) Lie algebra (see [7, 5]). Here we are interested in extending these results to the case where $R$ is a prime or semi-prime ring.
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paper_title: Lie structure of prime rings of characteristic 2

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In this paper the Lie structure of prime rings of characteristic 2 is discussed. Results on Lie ideals are obtained. These results are then applied to the group of units of the ring, and also to Lie ideals of the symmetric elements when the ring has an involution. This work extends recent results of I. N. Herstein, C. Lanski and T. S. Erickson on prime rings whose characteristic is not 2, and results of S. Montgomery on simple rings of characteristi c 2. 1* Prime rings* We first extend the results of Herstein [5]. Unless otherwise specified, all rings will be associative. If R is a ring, R has a Lie structure given by the product [x, y] = xy — yx, for x,yeR. A Lie ideal of R is any additive subgroup U of R with [u, r]e U for all u e U and reR. By a commutative Lie ideal we mean a Lie ideal which generates a commutative subring of R. Denote the center of R by Z. We recall that if R is prime, then the nonzero elements of Z are not zero divisors in R. In this case, if Z Φ 0 and F is the quotient field of R, then R ®ZF is a prime ring, every element of which can be written in the form r (x) α"1 for ae Z, a Φ 0. Thus R®ZF is naturally isomorphic to RZ~~\ the localization of R at Z. We will consider R imbedded in RZ~ι in the usual way (see [2]). We begin with some easy lemmas. LEMMA 1. If R is semi prime and U is a Lie ideal of R with u2 = 0 for all ue U, then U = 0.
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paper_title: Second cohomology of Lie rings and the Schur multiplier

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We exhibit an explicit construction for the second cohomology group $H^2(L, A)$ for a Lie ring $L$ and a trivial $L$-module $A$. We show how the elements of $H^2(L, A)$ correspond one-to-one to the equivalence classes of central extensions of $L$ by $A$, where $A$ now is considered as an abelian Lie ring. For a finite Lie ring $L$ we also show that $H^2(L, \C^*) \cong M(L)$, where $M(L)$ denotes the Schur multiplier of $L$. These results match precisely the analogue situation in group theory.
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paper_title: Lie algebra application to mobile robot control: a tutorial

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Lie algebra is an area of mathematics that is largely used by electrical engineer students, mainly at post-graduation level in the control area. The purpose of this paper is to illustrate the use of Lie algebra to control nonlinear systems, essentially in the framework of mobile robot control. The study of path following control of a mobile robot using an input-output feedback linearization controller is performed. The effectiveness of the nonlinear controller is illustrated with simulation examples.
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paper_title: Faithful Representations of Lie Algebras

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The object of this paper is to give a new proof of the theorem that every Lie algebra over a field K of characteristic zero, has a faithful representation. The first proof of this result, at least when K is algebraically closed, is due to Ado (1). Later Cartan (2) gave a simpler and entirely different proof for the case when K is the field of either real or complex numbers. Cartan's proof depends on the integration of the Maurer-Cartan equations and therefore is of a non-algebraic character.! The present proof is of course algebraic and seems to differ from the earlier ones in approaching the problem quite directly. Also the result established is slightly sharper than the usual one in so far as we assert the existence of a faithful representation in which every element of the maximal nilpotent ideal of the given Lie algebra is mapped on a nilpotent matrix. I am very much indebted to Professor C. Chevalley for his advice and help in improving the presentation of the proof. Also I should like to thank Dr. G. D. Mostow for many interesting and valuable discussions. All algebras (whether Lie algebras or associative algebras) and vector spaces appearing in this paper are to be understood over the basic field K. A linear Lie algebra 2 is a Lie algebra whose elements are endomorphisms of some given vector space, the bracket operation in 2 being defined by [X,Y] = XY YX. As far as possible we follow the notation and terminology of Chevalley's book (3) and his papers (4). In particular, if 2 is a Lie algebra and X e 2 we denote by ad X the derivation of 2 defined by (ad X)Y = [XY](Y ).2 The following notion of the semidirect sum of a Lie algebra and its algebra of derivations' is important for our purpose. DEFINITION. Let S be a Lie algebra and i) the algebra of its derivations. By the semidirect sum of V and Z is meant a Lie algebra 2 + ) defined as follows. Considered as a vector space 2 + Z is the direct sum of 2 and Z so that an element of 2 + i) is a pair (X, D) with X e and D e Z. The bracket operation in S + Z is defined by
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paper_title: Nilpotent groups admitting an almost regular automorphism of order four

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We consider locally nilpotent periodic groups admitting an almost regular automorphism of order 4. The following are results are proved: (1) If a locally nilpotent periodic group G admits an automorphism ϕ of order 4 having exactly m<∞ fixed points, then (a) the subgroup {ie176-1} contains a subgroup of m-bounded index in {ie176-2} which is nilpotent of m-bounded class, and (b) the group G contains a subgroup V of m-bounded index such that the subgroup {ie176-3} is nilpotent of m-bounded class (Theorem 1); (2) If a locally nilpotent periodic group G admits an automorphism ϕ of order 4 having exactly m<∞ fixed points, then it contains a subgroup V of m-bounded index such that, for some m-bounded number f(m), the subgroup {ie176-4}, generated by all f(m) th powers of elements in {ie176-5} is nilpotent of class ≤3 (Theorem 2).
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paper_title: On Almost Regular Automorphisms of Finite p-Groups

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Abstract In this paper we prove that there are functions f ( p , m , n ) and h ( m ) such that any finite p -group with an automorphism of order p n , whose centralizer has p m points, has a subgroup of derived length ⩽ h ( m ) and index ⩽ f ( p , m , n ). This result gives a positive answer to a problem raised by E. I. Khukhro (see also Problem 14.96 from the “Kourovka Notebook” (1999, E. I. Khukhro and V. D. Mazurov (Eds.), “The Kourovka Notebook: Unsolved Problems in Group Theory,” 14th ed., Novosibirsk)).
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paper_title: Introduction To Lie Algebras And Representation Theory

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Thank you very much for downloading introduction to lie algebras and representation theory. As you may know, people have search numerous times for their favorite books like this introduction to lie algebras and representation theory, but end up in harmful downloads. Rather than enjoying a good book with a cup of tea in the afternoon, instead they cope with some harmful virus inside their desktop computer.
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paper_title: Groups and Lie Algebras with Almost Regular Automorphisms

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Abstract It is proved that if a locally nilpotent group G admits an almost regular automorphism of prime order p then G contains a nilpotent subgroup G 1 such that | G : G 1 |≤ƒ( p , m ) and the class of nilpotency of G 1 ƒ g ( p ), where ƒ is a function on p and the number of fixed elements m and g depends on p only. An analog is proved for Lie rings (not necessarily locally nilpotent). These give an affirmative answer to the questions raised by Khukhro.
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paper_title: Finite groups and Lie rings with a metacyclic Frobenius group of automorphisms

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Suppose that a finite group $G$ admits a Frobenius group of automorphisms FH of coprime order with cyclic kernel F and complement H such that the fixed point subgroup $C_G(H)$ of the complement is nilpotent of class $c$. It is proved that $G$ has a nilpotent characteristic subgroup of index bounded in terms of $c$, $|C_G(F)|$, and $|FH|$ whose nilpotency class is bounded in terms of $c$ and $|H|$ only. This generalizes the previous theorem of the authors and P. Shumyatsky, where for the case of $C_G(F)=1$ the whole group was proved to be nilpotent of $(c,|H|)$-bounded class. Examples show that the condition of $F$ being cyclic is essential. B. Hartley's theorem based on the classification provides reduction to soluble groups. Then representation theory arguments are used to bound the index of the Fitting subgroup. Lie ring methods are used for nilpotent groups. A similar theorem on Lie rings with a metacyclic Frobenius group of automorphisms $FH$ is also proved.
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paper_title: Right alternative rings

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A number of papers have appeared with the purpose of generalizing Albert’s long-standing result [l, 21 that a simple right alternative algebra of finite dimension over a field of characteristic f2 that has a unit element e and an idempotent c # e is necessarily alternative. Until now results depended on rather strong additional assumptions such as other identities [S, 14, 17, 36, 371 or internal conditions on the algebra [9, 10, 12, 15, 27, 28, 301. Essential progress has been achieved by Micheev who showed that the identity (x, x, Y)~ = 0 holds in 2-torsion free right alternative algebras [27]. This paper starts collecting information on two natural concepts in a right alternative algebra R, the submodule M generated by all alternators (x, x, y), and a new nucleus N, . The later sections deal mainly with results on simple right alternative algebras. A simple 2-torsion free right alternative algebra is either alternative, hence associative or a Cayley algebra over its center, or the following statements hold:
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paper_title: The prime radical in alternative rings

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The characterization by J. Levitzki of the prime radical of an associative ring R as the set of strongly nilpotent elements of R is adapted here to apply to a wide class of nonassociative rings. As a consequence it is shown that the prime radical is a hereditary radical for the class of alternative rings and that the prime radical of an alternative ring coincides with the prime radical of its attached Jordan ring. In 1951 J. Levitzki characterized the prime radical of an associative ring R as the set of all elements r E R such that every m-sequence beginning with r ends in zero [3]. An m-sequence was defined to be a sequence {ao, a1,.... an ,... } such that ai C aij1 Rai_1 for i = 1, 2 .... Recently, C. Tsai [9] has given a similar characterization for the prime radical of a Jordan ring. Here we extend this characterization to the class of all s-rings and, as a consequence, are able to show that the prime radical is a hereditary radical on the class of alternative rings (i.e., if A is an ideal of an alternative ring R, then P(A) = A n P(R)) and that P(R) = P(R+) for all 2 and 3-torsion free alternative rings R. Although it is not known whether the prime radical is hereditary on the class of all s-rings, a partial result in this direction is obtained. Recall that a not necessarily associative ring R is called an s-ring for a positive integer s if As is an ideal of R whenever A is an ideal of R (As denotes the set of all sums of products a, a2 ... aS for ai E A under all possible associations). An ideal P of R is called a prime ideal if whenever A1 A2 ... As 5 P then Ai 5 P for some i. Here A1A2 ...As denotes the product of the ideals under all possible associations. The prime radical, P(R), of R is the intersection of all prime ideals of R and can be characterized as the set of all elements r E R such that every complementary system M of R which contains r also contains 0. A set M in R is a complementary system if whenever A1,A2, ...As ares ideals of R such thatAi n M# 0for i = 1, 2, ...,s, then (A1A2 ... As) n M # 0 [6], [8], [10]. To make this article self-contained we mention the following three properties of P(R) which hold for any s-ring R. Proofs can be found in [6], [8] and [10]. (a) P(R) = 0 if and only if R contains no nonzero nilpotent ideals. (b) P(R/P(R)) = 0. Received by the editors May 29, 1975. AMS (MOS) subject classifications (1970). Primary 17D05, 17E05.
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paper_title: Right alternative rings of characteristic two

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for all w, x, y and showed by example that (1.1) can fail to hold. Prior to this, Kleinfeld [1 ] generalized the Skornyakov theorem in another direction by assuming only the absence of one sort of nilpotent element. We now specify Kleinfeld's result in detail. Let F be the free nonassociative ring generated by xi and x2 and suppose that R is any right alternative ring. Kleinfeld calls t, u, v in R an alternative triple if (i) there exist elements a [xl, X2], 3 [Xl, x2], 7Y [xl, X2] in F and elements ri, r2 in R such that t =a [ri, r2 ], u =-1 [ri, r2], v = y [ri, r2] and (ii) if si and S2 are elements from an arbitrary alternative ring, and if t'=a [sl, S2], u'=3[sl, S2], v'-=Y[sl, S2], then (t', u', v') =0. The ring R is said to have property (P) if t, u, v an alternative triple in R and (t, u, V)2=0 imply (t, u, v) =0. By the definition of an alternative triple, an alternative ring has property (P). Kleinfeld's result is the converse, assuming characteristic not two; that is, a right alternative ring of characteristic not two is alternative if (and only if) it has property (P). We herein extend this line of investigation by proving that a right alternative ring of characteristic two, satisfying (1.1), is alternative if (and only if) it has property (P). The methods are mainly those used in [2], coupled with two essential lemmas (numbered 4 and 5 in our paper) due to Kleinfeld. Following [2], we say that R is strongly right alternative if R is a right alternative ring satisfying (1.1) . Throughout the paper, R will always denote such a ring, with the additional hypothesis that R have characteristic two.
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paper_title: Simple Alternative Rings

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One of the ways of gaining an insight into the nature of a class of rings is to determine all the simple ones. In the case of associative rings some restriction, such as the existence of maximal or minimal right ideals is usually made in order to characterize the simple ones, for otherwise one encounters seemingly pathological examples. The theory of simple alternative rings is therefore incongruous in that it presents no additional difficulties. More precisely, if one defines a ring to be simple provided it has no proper two-sided ideals and is not a nil ring then the main result of this paper may be stated as follows: A simple alternative ring is either a Cayley-Dickson algebra or associative. Thus it would seem that the distinction between alternative and associative rings is really insignificant. Besides the machinery developed in [3], a new identity plays a vital part in the argument. This identity asserts that in any alternative ring fourth powers of commutators associate with any pair of elements of the ring. In the original version of the author's argument this identity was proved with the additional assumption of simplicity. Thanks to R. H. Bruck, who modified that argument, the hypothesis of simplicity is now superfluous. Such an identity is likely to be a useful device in the study of general alternative rings. The next stepping stone is an adaptation of A. A. Albert's result [21, in the form of Theorem 2.8 of this paper. It is used to reduce the proof of the main theorem to a consideration of simple alternative rings in which the fourth power of every commutator is zero. Such rings have no nilpotent elements, from which one infers that they are fields.
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paper_title: RIGHT NUCLEUS IN GENERALIZED RIGHT ALTERNATIVE RINGS

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Some properties of the right nucleus in generalized right alternative rings have been presented in this paper. In a generalized right alternative ring R which is finitely generated or free of locally nilpotent ideals, the right nucleus Nr equals the center C. Also, if R is prime and Nr  C, then the associator ideal of R is locally nilpotent. Seong Nam [5] studied the properties of the right nucleus in right alternative algebra. He showed that if R is a prime right alternative algebra of char. ≠ 2 and Right nucleus Nr is not equal to the center C, then the associator ideal of R is locally nilpotent. But the problem arises when it come with the study of generalized right alternative ring as the ring dose not absorb the right alternative identity. In this paper we consider our ring to be generalized right alternative ring and try to prove the results of Seong Nam [5]. At the end of this paper we give an example to show that the generalized right alternative ring is not right alternative.
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paper_title: LEFT QUOTIENT RINGS OF ALTERNATIVE RINGS

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In this paper we develop a Fountain–Gould-like Goldie theory for alternative rings. We characterize alternative rings which are Fountain–Gould left orders in semiprime alternative rings coinciding with their socle, and those which are Fountain–Gould left orders in semiprime artinian alternative rings.
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paper_title: The Wedderburn principal theorem for alternative algebras

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for all a, x in 21. I t is clear that associative algebras are alternative. The most famous examples of alternative algebras which are not associative are the so-called Cayley-Dickson algebras of order 8 over $. Let S be an algebra of order 2 over % which is either a separable quadratic field over 5 or the direct sum 5 ©3There is one automorphism z—>z of S (over %) which is not the identity automorphism. The associative algebra O = 3~\~S with elements
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paper_title: Goldie’s theorem for alternative rings

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It is known that the socle of a semiprime Goldie ring is generated by a central idempotent and that a prime Goldie ring with a nonzero socle is a simple artinian ring. We prove the extension of these results to alternative rings. We also give an analogue of Goldie's theorem for alternative rings. A Goldielike theorem was obtained earlier by the authors for noetherian alternative rings by a quite different method.
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paper_title: Inner derivations of non-associative algebras

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A compression device for exerting pressure on an arm, shoulder, and/or trunk of a patient in need thereof (for example, a patient with hyperalgia or recovering from surgery in which the lymphatic system is affected), including an arm compression hose, a shoulder part for exerting pressure on the shoulder and trunk area, and a band-shaped fastening means for positioning the shoulder part and exerting pressure on the shoulder part. The arm compression hose exerts a pressure that decreases from a maximum pressure at the wrist or hand to a minimum pressure near the shoulder end of the arm, where the minimum pressure is approximately 70% of the maximum pressure. One or more lining pockets can be constructed on the inner lining of the compression device, where each lining pocket can hold one or more compression pads to increase tissue pressure in one or more body areas in need thereof. The compression pads each can have a shape that approximately conforms to the shape of the body part to which it is applied. The shoulder part can also have a shape that approximately conforms to the contour of the shoulder/trunk area to which it is applied. In addition, compression pants can be prepared with lining pockets for receiving compression pads. In one embodiment, compression pants include one or more donut-shaped pads or equivalents thereof that are placed in one or more lining pockets, each of which surrounds one or more osteoma openings.
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paper_title: Alternative regular rings without nilpotent elements

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An energy efficient general purpose injection molding employs a hydraulic drive for injection and clamp machine functions and an electric drive for screw recovery. Both drives use AC "squirrel cage" induction motors under vector control. Speed command signals for the vector controls are generated by the machine's controller utilizing state transition and predictive signal techniques to account for motor and motor/pump response latencies. Hydraulic drive efficiency is improved by varying motor speed/pump output to match cycle requirements to retain hydraulic drive advantages for mold clamp and injection functions while improving the electric drive performance for screw recovery.
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paper_title: The Radical of an Alternative Ring

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In this paper we shall show that N. Jacobson's definition of the radical of an (associative) ring [J1, B] applies to alternative rings [Z1, M], and we shall develop some of the elementary properties of this radical. The radical of an alternative ring was first discussed by M. Zorn [Z2] under certain finiteness assumptions. Dubisch and Perlis [D P] have more recently studied the radical of an alternative algebra (of finite order). We make no finiteness assumptions. We do, however, prove that the chain conditions employed by Zorn [Z2, (4.2.1)(4.2.3)] ensure that the radical defined by him coincides with that defined in this paper. Our discussion applies equally well to algebras of possibly infinite order [JI, ?6] so that our results essentially contain some of the results of Dubisch and Perlis. We have been unable to discover the relation between the radical and maximal right ideals, nor have we developed any parallel for Jacobson's structure theory of associative rings. The enlarged radical of Brown and McCoy [B-McC] will yield a type of structure theory for arbitrary non-associative rings, but because of the generality involved we prefer to leave a discussion of this interesting fact to a subsequent publication.
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paper_title: The Structure of Alternative Division Rings

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A small reed switch having a glass tubularly shaped envelope containing a pair of reed contacts is modified to ensure its being biased to either an open or a closed position. A donut-shaped piece of a rubber-bonded, barium-ferrite, magnetic material is circumferentially mounted on the reed switch at a position where its magnetic field influences the contacts to an open position or a closed position. Thusly arranged, an actuating magnet, having a sufficient field at a single pre-established distance from the contacts, actuates the switch from all radial directions from the switch. Potting the modified switch in an epoxy resin further ensures a greater reliability and makes it ideal for implantations in laboratory animals.
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paper_title: Jordan homomorphisms and right alternative rings

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for every a, bER. R. L. San Soucie [4] calls a nonassociative ring R strongly right alternative in case its right multiplications a': x-*xa satisfy (I) and (II). (Every right alternative ring in which 2a=0 implies a=0 is strongly right alternative just as (I) implies (II) under the same assumption in the associative case.) In a recent paper [5] we developed some identities for Jordan homomorphisms in the associative case which were extensions of those previously given. (See, especially, I. N. Herstein [1].) It turns out that the nonassociative analogues of these identities are useful in reducing strongly right alternative rings to alternative rings. To do this one invokes the property (P0) If x, y, zER and (x, y, Z)2=0, then (x, y, z) =0 provided x has one of the forms y, [y, z], [y, z]y, (y, y, z) or provided z=wy and x= (y, y, w). (Property (P0) is the operational form of E. Kleinfeld's Property (P) [3].) We give in this note a proof that (I), (II) and (P0) imply that R is an alternative ring. This result subsumes those of Kleinfeld [3] and of San Soucie [4] in this connection. Our main interest is not in the slightly greater generality of this result but rather in the method of proof. The reader will find, we hope, that our proof is straightforward and relatively brief. We have made our presentation self-contained but we have relegated some simple computations, most of which are strictly analogous to those given by us in [5], to an Appendix.
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paper_title: Theorie der alternativen ringe

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A new catalyst component and its use with an organoaluminum compound, which component is a brown solid of high surface area and large pore volume comprising beta titanium trichloride and a small amount of an organic electron pair donor compound. This solid when used in conjunction with an organoaluminum compound to polymerize alpha-olefins produces product polymer at substantially increased rates and yields compared to present commercial, purple titanium trichloride while coproducing reduced amounts of low-molecular-weight and, particularly, amorphous polymer. Combinations of this new catalyst component and an organoaluminum compound can be further improved in their catalytic properties by addition of small amounts of modifiers, alone and in combination. Such combinations with or without modifiers show good sensitivity to hydrogen used as a molecular weight controlling agent. The combinations are useful for slurry, bulk and vapor phase polymerization of alpha-olefins such as propylene.
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paper_title: On the commutativity of certain rings

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An ink jet printer includes apparatus for preventing ink clogs from interfering with the flow of ink from a printing nozzle during a printing operation. The ink-declogging apparatus includes a vacuum pump having a vacuum chamber and a member movable with respect to such chamber to adjust the pressure therein. A printing nozzle is operably coupled to the chamber when the printing nozzle is not being used in a print operation. A stepper motor controls the position of the movable member relative to the chamber to selectively provide at least two different preset levels of vacuum (suction) to the printing nozzle. Having the capability of controlling the level of vacuum applied to the printing nozzle, a high vacuum need only be applied in situations warranting its use (e.g. to remove ink clogs), and the waste of ink (by an unnecessarily high vacuum) can be avoided.
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paper_title: On a class of right alternative rings without nilpotent ideals

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THEOREM. A simple right alternative ring R which is not associative is alternative if and only if R has an idempotent e sub that there are no nilpotent elements in R,(e) and R,(e). Because simple alternative rings that are not associative are known to be Cayley-Dickson algebras [5] one easily sees that the condition is necessary. Most of the paper is taken up with establishing that it is also sufficient. In the process we prove the following more general result.
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paper_title: Derivable Maps on Alternative Rings

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Let D be a mapping from an alternative ring R into itself satisfying D(ab) = D(a)b + aD(b) for all a; b 2 R. Under some conditions on R, we show that D is additive.
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paper_title: THE MAXIMAL LEFT QUOTIENT RINGS OF ALTERNATIVE RINGS

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ABSTRACT We study the notion of a (general) left quotient ring of an alternative ring and show the existence of a maximal left quotient ring for every alternative ring that is a left quotient ring of itself.
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paper_title: Nilpotent right alternative rings and bol circle loops

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We determine the nilpotent right alternative rings of prime power oirder pn n ≥ 4, which are not left alternative. Those which are strongly right alternative become Bol loops under the circle operation. The smallest Bol circle loop has order 16. There are six such loops, all of which appear to be new.
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paper_title: RIGHT NUCLEUS IN RIGHT ALTERNATIVE RINGS

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I n this paper we prove that if R is a semiprime and purely non-associative right alternative ring, then N = C. Also we show that the right nucleus N r = C if R is purely non-associative provided that either R has no locally nilpotent ideals or R is semi prime and finitely generated mod N r
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paper_title: On free alternative rings

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An example is presented that ansevers in the negative the question of whether the square of every commutator need always lie in the nucleus. Also, we show the existence of specific nilpotent elements in the free alternative ring on four or more generators, and prove abstractly the existence of an ideal I≠0, and I2=0.
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paper_title: Right Alternative Rings with Idempotents

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for all elements x and a. Throughout this paper, all rings will be assumed to have characteristic prime to 2 and 3. If a right alternative ring A has an idempotent, it has an Albert [l] decomposition A = A,(e) + A,,,(e) + Ao(e), where A,(e) and A,(e) are closed under Jordan product. We show that if A has a decomposition where A,(e)+ and A,(e)+ are simple Jordan algebras and A satisfies a few other conditions, then A is alternative or A = A,(e) @ A,,(e) (direct sum). The Albert decomposition with respect to an idempotent e is useful in right alternative rings provided it can be stretched to a Peirce decomposition. This means it is necessary for (e, e, A) = 0. We simply assume our idempotent has this property. Our theorem is:
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paper_title: On Quotient Rings in Alternative Rings

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We introduce a notion of left nonsingularity for alternative rings and prove that an alternative ring is left nonsingular if and only if every essential left ideal is dense, if and only if its maximal left quotient ring is von Neumann regular (a Johnson-like Theorem). Finally, we obtain a Gabriel-like Theorem for alternative rings.
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paper_title: Rings With Minimal Condition for Left Ideals

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One finds in the literature a thorough-going discussion of rings without radical and with minimal condition for left ideals (semi-simple rings). For the structure of a ring whose quotient-ring with respect to the radical is semi-simple one can refer to the investigations of K6the (see K below). In this paper we shall examine the structure of a ring A with radical R D 0 and with minimal condition for left ideals ("general" MLI ring). The key-stone of our investigations is the fact that the radical of A is nilpotent, and this result we shall establish in ?1. In ?2 we shall prove that the sum of all minimal non-zero left ideals is a completely-reducible left ideal 9A, and in ?3 we shall examine the distribution of idempotent and nilpotent left ideals in 9)1. In ??4-6 we shall discuss the two "extreme cases": (1) when A is nilpotent, and (2) when A is idempotent. For a non-nilpotent A we shall prove that the existence of either a right-hand or a left-hand identity is sufficient for the existence of a composition series of left ideals of A. If A is any MLI ring, one can find a smallest exponent k such that Ak = Ak+l = ... . In ?7 we show that A is the sum of Ak (which is idempotent) and a nilpotent MLI ring. We wish to emphasize the fact that A is to be regarded throughout as a ring without operators. In ?8, however, we shall see that some of our most interesting results are valid for operator domains of a certain type. We conclude the Introduction with an explanation of our notation and terminology. Rings and subrings will usually be denoted by roman capitals; we shall use gothic letters when it is desirable to emphasize the fact that a subring is an ideal. By the statement "a is a left (right) ideal of A" we shall mean that a is an additive abelian group which admits the elements of A as left-hand (right-hand) operators. Observe that our definition does not' imply that a is a subring of A. The term "left ideal," with no qualifying phrase, will always mean "left ideal of the basic ring." A ring with minimal condition for left (right) ideals which are contained in itself will be called an MLI (MRI) ring. Finally we point out that if a and b are subrings of A, then [a, b] denotes the cross-cut of a and b, while (a, b) represents the compound (join) of a and b-i.e.
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paper_title: A Taste of Jordan Algebras

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In this book, Kevin McCrimmon describes the history of Jordan Algebras and he describes in full mathematical detail the recent structure theory for Jordan algebras of arbitrary dimension due to Efim Zel'manov. To keep the exposition elementary, the structure theory is developed for linear Jordan algebras, though the modern quadratic methods are used throughout. Both the quadratic methods and the Zelmanov results go beyond the previous textbooks on Jordan theory, written in the 1960's and 1980's before the theory reached its final form. ::: ::: This book is intended for graduate students and for individuals wishing to learn more about Jordan algebras. No previous knowledge is required beyond the standard first-year graduate algebra course. General students of algebra can profit from exposure to nonassociative algebras, and students or professional mathematicians working in areas such as Lie algebras, differential geometry, functional analysis, or exceptional groups and geometry can also profit from acquaintance with the material. Jordan algebras crop up in many surprising settings and can be applied to a variety of mathematical areas. ::: ::: Kevin McCrimmon introduced the concept of a quadratic Jordan algebra and developed a structure theory of Jordan algebras over an arbitrary ring of scalars. He is a Professor of Mathematics at the University of Virginia and the author of more than 100 research papers.
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paper_title: Hereditary radicals in Jordan rings

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The object of this paper is to examine some radical properties of quadratic Jordan algebras and to show that under certain conditions, R(3) = 3 nR(3) where Q3 is an ideal of a quadratic Jordan algebra 3, R(3) is the radical of Q3, and R(3) is the radical of 3. 1. Preliminaries. We adopt the notation and terminology of an earlier paper [2] concerning quadratic Jordan algebras (defined by the quadratic operators UJ) as opposed to linear Jordan algebras (defined by the linear operators Lx). Thus we have a product U,y linear in y and quadratic in x satisfying the following axioms as well as their linearizations: (UQJ I) U1=I (1 the unit); (UQJ II) U (u)y = Ux U U x; (UQJ III) UXV',X =VX ,UX (VXgz={xyz}1=UX,y). Throughout this paper 3 will denote a quadratic Jordan algebra over an arbitrary ring of scalars (D. Define a property R of a class of rings (e.g. associative rings or Jordan rings) to be a radical property it it satisfies the following three conditions [1]: (a) Every homomorphic image of an R ring is again an R ring. (b) Every ring 3 contains an R ideal R(13) which contains every other R ideal of 3. The maximal R ideal R(Q3) is called the R radical of 3. (c) For Q3 an ideal of 3, if Q3 and 3/Q3 are R rings, then so is 3. An immediate consequence of this definition is R(Q/R(Q))=O. If R(f=O, 0Q3 is said to be R semisimple. Many well-known radical properties, but not all, satisfy a further condition: (d) Every ideal of an R ring is again an R ring (i.e. property R is inherited by ideals of an R ring). If a radical property satisfies condition (d) that property is called hereditary. Received by the editors August 2, 1971. AMS 1969 subject classifications. Primary 1740.
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paper_title: Jordan homomorphisms of rings

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The primary aim of this paper is to study mappings J of rings that are additive and that satisfy the conditions ::: ::: $$ {\left( {{a^2}} \right)^J} = {\left( {{a^J}} \right)^2},\;{\left( {aba} \right)^J} = {a^J}{b^J}{a^J} $$ ::: ::: (1) ::: ::: Such mappings will be called Jordan homomorphisms. If the additive groups admit the operator 1/2 in the sense that 2x = a has a unique solution (1/2)a for every a, then conditions (1) are equivalent to the simpler condition ::: ::: $$ {\left( {ab} \right)^J} + {\left( {ba} \right)^J} = {a^J}{b^J} + {b^J}{a^J} $$ ::: ::: (2) ::: ::: Mappings satisfying (2) were first considered by Ancochea [1], [2](1). The modification to (1) is essentially due to Kaplansky [13]. Its purpose is to obviate the necessity of imposing any restriction on the additive groups of the rings under consideration.
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paper_title: Structure and Representations of Jordan Algebras

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A conveyor mechanism for automatically removing articles, such as cheeses, floating in a liquid includes a horizontal section adapted to be positioned below the surface of the liquid and an integral upwardly inclined section. Articles that float upon the horizontal section are moved to the inclined section which lifts them out of the liquid and delivers them to an elevated point. The articles are prevented from jamming the conveyor by moving stepped walls located on either side of the horizontal section at the location where jamming is likely to occur. The stepped walls move longitudinally back and forth 180 DEG out of phase with one another to gently bump and align the articles. In the preferred embodiment, a single motor drives the conveyor and rotates cam wheels which move the stepped walls.
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paper_title: Non-associative algebra and its applications

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This volume contains the proceedings of the Third International Conference on Non-Associative Algebra and Its Applications, held in Oviedo, Spain, July 12-17, 1993. The conference brought together specialists from all over the world who work in this field. All aspects of non-associative algebra are covered. Topics range from purely mathematical subjects to a wide spectrum of applications, and from state-of-the-art articles to overview papers. This collection should point the way for further research. The volume should be of interest to researchers in mathematics as well as those whose work involves the application of non-associative algebra in such areas as physics, biology and genetics.
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paper_title: Homomorphisms of Jordan rings of self-adjoint elements

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In a previous paper [4](1) we have defined a special Jordan ring to be a a subset of an associative ring which is a subgroup of the additive group and which is closed under the compositions a→a 2and (a, b)→aba. Such systems are also closed under the compositions (a, b) → ab+ba= {a, b} and (a, b, c) → abc+cba. The simplest instances of special Jordan rings are the associative rings themselves. In our previous paper we studied the (Jordan) homomorphisms of these rings. These are the mappings J of associative rings such that ::: ::: $$ {\left( {a + b} \right)^J} = {a^J} + {b^J},\;{\left( {{a^2}} \right)^J} = {\left( {{a^J}} \right)^2},\;{\left( {aba} \right)^J} = {a^J}{b^J}{a^J} $$ ::: ::: (1) ::: ::: A second important class of special Jordan rings is obtained as follows. Let \( H \) be an associative ring with an involution a → a *, that is, a mapping a→a * such that ::: ::: $$ {\left( {a + b} \right)^*} = {a^*} + {b^*},\;{\left( {ab} \right)^*} = {b^*}{a^*},\;{a^{**}} = a $$ ::: ::: (2) ::: ::: Let \( H \) denote the set of self-adjoint elements h = h *. Then is a special Jordan ring. In this paper we shall study the homomorphisms of the rings of this type. It is noteworthy that the Jordan rings of this type include those of our former paper(2).
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paper_title: A general theory of jordan rings.

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Herbicidal cyanoalkanapyrazoles of the formula: (I) where N IS 3, 4, OR 5; Q is -O-CH3 or -S(O)m-CH3; where m is 0, 1 or 2; and R1 is hydrogen or methyl; V is hydrogen, fluorine or chlorine; X is fluorine, chlorine, bromine, iodine, cyano or methoxy; Y is hydrogen, fluorine, or chlorine; and Z is hydrogen or fluorine; PROVIDED THAT:
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paper_title: THE LEVITZKI RADICAL IN JORDAN RINGS

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The main purpose of this paper is to give an external characterization of the Levitzki radical of a Jordan ring 2f as the intersection of a family of prime ideals W. This characterization coincides with that of associative rings which was given by Babic in [1I]. Applying this characterization, it is easy to see that the Levitzki radical of a Jordan ring contains the prime radical of the same ring. For associative rings the same statement is well known, since the prime radical in associative rings is called the Baer radical. If the minimal condition on ideals holds on Jordan ring 2, then the Levitzki radical, L(2f), and the prime radical, R(2f) of 2f coincide. Throughout this paper, any Jordan ring 2f, that is a (nonassociative) ring satisfying (1) ab = ba, and (2) a2(ab) =a(a2b) for all a, b in 2t, and any of its subrings satisfy the conditions, (3) 2a = 0 implies a = 0 and (4) if a is in a subring C of 2 then there exists a unique element x in C such that 2x = a. In a Jordan ring, the following identity (*) is well known. One can find the proof in [3 ].
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paper_title: Jordan structures in mathematics and physics

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The aim of this paper is to offer an overview of the most important applications of Jordan structures inside mathematics and also to physics, up-dated references being included. For a more detailed treatment of this topic see - especially - the recent book Iordanescu [364w], where sugestions for further developments are given through many open problems, comments and remarks pointed out throughout the text. ::: Nowadays, mathematics becomes more and more nonassociative and my prediction is that in few years nonassociativity will govern mathematics and applied sciences. ::: Keywords: Jordan algebra, Jordan triple system, Jordan pair, JB-, JB*-, JBW-, JBW*-, JH*-algebra, Ricatti equation, Riemann space, symmetric space, R-space, octonion plane, projective plane, Barbilian space, Tzitzeica equation, quantum group, B\"acklund-Darboux transformation, Hopf algebra, Yang-Baxter equation, KP equation, Sato Grassmann manifold, genetic algebra, random quadratic form.
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paper_title: Units in Integral Loop Rings

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Abstract G. Higman has proved a classical result giving necessary and sufficient conditions for the units of an integral group ring to be trivial. In this paper we extend this result to loop rings of some diassociative loops.
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paper_title: Units in Alternative Integral Loop Rings

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We prove the isomorphism problem for integral loop rings of finitely generated RA loops using a decomposition of the loop of units. Also we describe the finitely generated RA loops whose loops of units satisfy a certain property.
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paper_title: Some results in the theory of linear non-associative algebras

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A waterproof pressure sensitive adhesive laminate is provided in which a flexible plastics backing sheet is coated with a bituminous adhesive composition containing a minor proportion of rubber or thermoplastic polymer. The backing sheet is reinforced with a mesh or a woven or non-woven fabric which is embedded in the sheet and provides substantial resistance to stretching.
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paper_title: Alternative Loop Rings with Solvable Unit Loops

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Let L be an RA loop, that is, a loop whose loop ring in any characteristic is an alternative, but not associative, ring. We find necessary and sufficient conditions for the (Moufang) unit loop of RL to be solvable when R is the ring of rational integers or an arbitrary field.
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paper_title: The Units of Group-Rings

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A shaped charge liner is made by forming a pair of hollow conical sub-liners, without any uncontrolled residual tangential shear stress (e.g. by a deep drawing method) so that they mate together to form a single conical liner. One sub-liner is inserted in the other and then one pair of subliner ends are locked together. Thereafter, the sub-liners are counter-rotated about their attachment point and locked together at their other pair of ends to retain the counter-rotation. This counter rotation generates opposite residual tangential shear stresses in the two sub-liners which may or may not be equal to each other.
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paper_title: Jordan decompositions in alternative loop rings

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It is observed that the additive as well as multiplicative Jordan decompositions hold in alternative loop algebras of finiteRA loops and theRA loops for which the additive Jordan decomposition holds in the integral loop ring are characterized. Multiplicative Jordan decomposition (MJD) inZL, whereL is a finiteRA loop with cyclic centre is analysed, besides settling MJD for integral loop rings of allRA loops of order ≤32. It is also shown that for any finiteRA loopL,U (ZL) is an almost splittable Moufang loop.
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paper_title: Alternative Loop Rings

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The right alternative law implies the left alternative law in loop rings of characteristic other than 2. We also exhibit a loop which fails to be a right Bol loop, even though its characteristic 2 loop rings are right alternative.
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paper_title: LOOP RINGS SATISFYING IDENTITIES OF BOL–MOUFANG TYPE

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The existence of loop rings that are not associative but which satisfy the Moufang or Bol identities is well known. Here we complete work started 25 years ago by establishing the existence of loop rings that satisfy any identity of "Bol–Moufang" type (without being associative). As it turns out, with one exception, loop rings satisfying an identity of Bol–Moufang type all satisfy a Moufang or Bol identity. We also highlight some similarities and differences in the consequences of several Bol–Moufang identities as they apply to loops and rings.
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paper_title: NORMALITY OF f-UNITARY UNITS IN AN ALTERNATIVE LOOP RING

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Let L be an RA loop, that is, a loop whose loop ring in any characteristic is an alternative, but not associative, ring. Let f : L → {±1} be a homomorphism and for α = ∑αll in the integral loop ring ZL, define αf = ∑αlf(l)l-1. A unit u ∈ ZL is said to be f-unitary if uf = ±u-1. The set of all f-unitary units is a subloop of , the loop of all units in ZL. In this paper, we find necessary and sufficient conditions for to be normal in .
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paper_title: A Construction of Loops which admit Right Alternative Loop Rings

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In this note, the authors offer a specific construction of loops whose loop rings are right, but not left, alternative.
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paper_title: Loops whose loop rings in characteristic 2 are alternative

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In this paper, the authors continue their investigation of loops which give rise to alternative loop rings. If the coefficient ring has characteristic 2, these loops turn out to form a surprisingly wide class, in contrast to the situation of characteristic ≠ 2. This paper describes many properties of this class, includes diverse examples of Moufang loops which are united by the fact that they have loop rings which are alternative, and discusses analogues in loop theory of a number of important group theoretic constructions.
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paper_title: Normal Subloops in the Integral Loop Ring of an $\RA$ Loop

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Disclosed herein is an exhaust system for a two-cycle internal combustion engine including a rotatable crankshaft, first and second cylinders firing 180 DEG apart, and first and second exhaust ports communicating with the first and second cylinders respectively, the exhaust system comprising a substantially Y-shaped hollow exhaust pipe having first, second and third branches each having an open end and each being substantially equal in length to the distance an acoustical wave will travel through the exhaust pipe during an interval over which the crankshaft rotates through substantially ten to twenty degrees of rotation at a predetermined engine speed, the open ends of the first and second branches being adapted to be coupled to the exhaust ports of the first and second cylinders respectively.
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paper_title: Units in alternative loop rings

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In this paper, we show that certain well known theorems concerning units in integral group rings hold more generally for integral loop rings which are alternative.
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paper_title: SRAR loops with more than two commutators

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Abstract Possession of a unique nonidentity commutator/associator is a property that dominates the theory of loops whose loop rings, while not associative, nevertheless satisfy an “interesting” identity. For instance, until now, all loops with loop rings satisfying the right Bol identity (such loops are called SRAR) have been known to have this property. In this paper, we present various constructions of other kinds of SRAR loops.
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paper_title: Alternative loop rings

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Contents. Preface. Introduction. I. Alternative Rings. Fundamentals. The real quaternions and the Cayley numbers. Generalized quaternion and Cayley-Dickson algebras. Composition algebras. Tensor products. II. An Introduction to Loop Theory and to Moufang Loops. What is a loop? Inverse property loops. Moufang loops. Hamiltonian loops. Examples of Moufang loops. III. Nonassociative Loop Rings. Loop rings. Alternative loop rings. The LC property. The nucleus and centre. The norm and trace. IV. RA Loops. Basic properties of RA loops. RA loops have LC. A description of an RA loop. V. The Classification of Finite RA loops. Reduction to indecomposables. Finite indecomposable groups. Finite indecomposable RA loops. Finite RA loops of small order. VI. The Jacobson and Prime Radicals. Augmentation ideals. Radicals of abelian group rings. Radicals of loop rings. The structure of a semisimple alternative algebra. VII. Loop Algebras of Finite Indecomposable RA Loops. Primitive idempotents of commutative rational group algebras. Rational loop algebras of finite RA loops. VIII. Units in Integral Loop Rings. Trivial torsion units. Bicyclic and Bass cyclic units. Trivial units. Trivial central units. Free subgroups. IX. Isomorphisms of Integral Alternative Loop Rings. The isomorphism theorem. Inner automorphisms of alternative algebras. Automorphisms of alternative loop algebras. Some conjectures of H.J. Zassenhaus. X. Isomorphisms of Commutative Group Algebras. Some results on tensor products of fields. Semisimple abelian group algebras. Modular group algebras of abelian groups. The equivalence problem. XI. Isomorphisms of Loop Algebras of Finite RA Loops. Semisimple loop algebras. Rational loop algebras. The equivalence problem. XII. Loops of Units. Reduction to torsion loops. Group identities. The centre of the unit loop. Describing large subgroups. Examples. XIII. Idempotents and Finite Conjugacy. Central idempotents. Nilpotent elements. Finite conjugacy. Bibliography. Index. Notation.
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paper_title: Isomorphisms of integral alternative loop rings

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This paper first settles the “isomorphism problem” for alternative loop rings; namely, it is shown that a Moufang loop whose integral loop ring is alternative is determined up to isomorphism by that loop ring. Secondly, it is shown that every normalized automorphism of an alternative loop ringZ L is the product of an inner automorphism ofQ L and an authomorphism ofL.
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paper_title: A class of loops with right alternative loop rings

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The purpose of this paper is to exhibit a class of loops which have strongly right alternative loop rings that are not alternative..
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paper_title: FINITE SUBLOOPS OF UNITS IN AN ALTERNATIVE LOOP RING

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An RA loop is a loop whose loop rings, in characteristic different from 2, are alternative but not associative. In this paper, we show that every finite subloop H of normalized units in the integral loop ring of an RA loop L is isomorphic to a subloop of L. Moreover, we show that there exist units -yi in the rational loop algebra QL such that y ( ( 1 (.. 1 'H-l)'Y2) ... )-y-Yk C L. Thus, a conjecture of Zassenhaus which is open for group rings holds for alternative loop rings (which are not associative).
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paper_title: A theorem on commutative power associative loop algebras

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Let L be a loop, written multiplicatively, and F an arbitrary field. Define multiplication in the vector space A, of all formal sums of a finite number of elements in L with coefficients in F, by the use of both distributive laws and the definition of multiplication in L. The resulting loop algebra A(L) over F is a linear nonassociative algebra (associative, if and only if L is a group). An algebra A is said to be power associative if the subalgebra FI[x] generated by an element x is an associative algebra for every x of A.
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paper_title: A BRIEF HISTORY OF LOOP RINGS

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Given a loopL and a commutative associative ringR with 1, one forms the loop ring RL just as one would form a group ring if L were a group. The theory of group rings has a long and rich history. In this paper, we sketch the history of loop rings which are not associative from early results of R. H. Bruck and L. J. Paige through the more recent discovery of alternative and right alternative rings and the work of O. Chein, D. A. Robinson and the author. 1. Origins Denition 1.1. A loop is an algebraic structureNL; O with a two-sided identity element such
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paper_title: TORSION UNITS IN ALTERNATIVE LOOP RINGS

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Let ZL denote the integral alternative loop ring of a finite loop L. If L is an abelian group, a well-known result of G. Higman says that ±g,g € L are the only torsion units (invertible elements of finite order) in ZL . When L is not abelian, another obvious source of units is the set ±y~l gy of conjugates of elements of L by invertible elements in the rational loop algebra QL . H. Zassenhaus has conjectured that all the torsion units in an integral group ring are of this form. In the alternative but not associative case, one can form potentially more torsion units by considering conjugates of conjugates V^\y7(g7l)V\ and so forth. In this paper we prove that every torsion unit in an alternative loop ring over Z is ± a conjugate of a conjugate of a loop element.
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paper_title: Congruences on left almost rings

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The aim of this paper is to characterize left almost rings by congrunces. We show that each homomophism of left amost rings defines a congrucne relation on left almost rings. We then discuss quotient left almiost rings. At the end we prove analogues of the ismorphism theorem for left almost rings.
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paper_title: On Left Primary and Weakly Left Primary Ideals in LA- Rings

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In this paper, we study left ideals, left primary and weakly left primary ideals in LA-rings. Some characterizations of left primary and weakly left primary ideals are obtained. Moreover, we investigate relationships left primary and weakly left primary ideals in LA-rings. Finally, we obtain necessary and sufficient conditions of a weakly left primary ideal to be a left primary ideals in LA- rings.
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paper_title: QUASI-IDEALS OF A $P$-REGULAR NEAR LEFT ALMOST RINGS

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S.J. Choi, P. Dheena and S. Manivasan studied property of quasi- ideals of P-regular nearring. In this page we study property of quasi-ideals of P-regular nLA-ring.
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paper_title: Application of Soft Sets to Non-Associative Rings

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Molodtsov developed the theory of soft sets which can be seen as an effective tool to deal with uncertainties. Since the introduction of this concept, the application of soft sets has been restricted to associative algebraic structures (groups, semigroups, associative rings, semi-rings etc). Acceptably, though the study of soft sets, where the base set of parameters is a commutative structure, has attracted the attention of many researchers for more than one decade. But on the other hand there are many sets which are naturally endowed by two compatible binary operations forming a non-associative ring and we may dig out examples which investigate a non-associative structure in the context of soft sets. Thus it seems natural to apply the concept of soft sets to non-commutative and non-associative structures. In present paper, we make a new approach to apply Molodtsov’s notion of soft sets to LA-ring (a class of non-associative ring). We extend the study of soft commutative rings from theoretical aspect.
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paper_title: On Near Left Almost Rings

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In this paper we give the notion of near left almost ring (ab- breviated as nLA-ring) (R, +, ·), i.e. (R, +) is an LA-group, (R, ·) is an LA- semigroup and one distributive property of '·' over '+' holds, where both the binary operations "+" and "·" are non-associative. An nLA-ring is a general- ization of an LA-ring and footed parallel to the near ring. Mathematics Subject Classification: 16A76, 20M25, 20N02
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Title: Literature Survey on Non-Associative Rings and Developments

Section 1: Introduction
Description 1: Provide an overview of the subject of non-associative rings, including historical context, key definitions, and key developments leading up to the creation of non-associative ring theory.

Section 2: Historical Perspective and Developments
Description 2: Describe the literature and developments of non-associative rings from different decades, highlighting key researchers and their contributions.

Section 3: Octonions
Description 3: Discuss the origin and significance of octonions as the first example of non-associative rings, including key properties and historical context.

Section 4: Lie Rings (1870-2015)
Description 4: Review the evolution and development of Lie rings, major contributions, and their applications in mathematics and physics.

Section 5: Alternative Rings (1930-2015)
Description 5: Survey the discovery and study of alternative rings, key properties, and significant results obtained by various researchers over the decades.

Section 6: Jordan Rings (1933-2011)
Description 6: Analyze the introduction and development of Jordan rings, their main properties, and their relevance to both mathematics and physics.

Section 7: Loop Rings (1944-2015)
Description 7: Examine the concept of loop rings, their development over time, and the various mathematical properties and applications identified by researchers.

Section 8: LA-Ring (2006-2016)
Description 8: Discuss the introduction of LA-rings, their unique properties, and the major developments in their study and applications within non-associative ring theory.

Section 9: Conclusions
Description 9: Summarize the overall importance of non-associative rings in mathematics and applied sciences, highlighting future directions and potential areas of research.