For 10 points, identify these algebraic structures, of which the integers are an example, which have addition and multiplication but do not require multiplicative inverses.
Modules are distinguished from vector spaces because their scalar coefficients come from one of these structures. If a certain type of descending chain in one of these things terminates, then it is referred to as Artinian. If one of these things is a PID, then all ideals are generated by only one element. The set of polynomials with real coefficients is referred to as one of these things, because the product and sum of two polynomials is itself a polynomial. Real square matrices of a given dimension form a non-commutative example of these things. For 10 points, identify these algebraic structures, of which the integers are an example, which have addition and multiplication but do not require multiplicative inverses.
Modules are distinguished from vector spaces because their scalar coefficients come from one of these structures.
commutative rings [anti-prompt on integral domain, unique factorization domain, UFD, principal ideal domain, PID, or Euclidean domain; do not accept or prompt on "field"] 