If the product of any two elements of a ring is itself an element of that ring, then that ring's one of these is said to be prime, whereas the smallest one is considered maximal.
Tsang showed that invertible content examples of this construct characterize Gaussian polynomials. Bézout names domains for which finitely generated ones imply principally generated ones. The maximum number of strict inclusions of them in certain groups is given by the Krull Dimension. (*) Buchberger's algorithm finds bases for one of these sets; that output is known as the Gröbner basis. For Prüfer varieties, every finitely generated one is invertible. Artinian rings add a decreasing condition on chains of these to an increasing condition that is the subject of Hilbert's Basis Theorem and is named for Noether. If the product of any two elements of a ring is itself an element of that ring, then that ring's one of these is said to be prime, whereas the smallest one is considered maximal. For 10 points, identify these subgroups, named by Kummer, that are invariant under multiplication and central to commutative ring theory.
Tsang showed that invertible content examples of this construct characterize Gaussian polynomials.
{Ideals}