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An associative ring, also called a Hecke ring, which has a technical definition in terms of commensurable subgroups.

A more common way to describe a Euclidean ring.

An extension ring R subset= S such that every element of S is integral over R.

There are at least two statements known as Schur's lemma. 1. The endomorphism ring of an irreducible module is a division algebra. 2. Let V, W be irreducible (linear) ...

The ring of integers of a number field K, denoted O_K, is the set of algebraic integers in K, which is a ring of dimension d over Z, where d is the extension degree of K over ...

An element B of a ring is nilpotent if there exists a positive integer k for which B^k=0.

If R is a ring (commutative with 1), the height of a prime ideal p is defined as the supremum of all n so that there is a chain p_0 subset ...p_(n-1) subset p_n=p where all ...

An extension A subset B of a group, ring, module, field, etc., such that A!=B.

The integral closure of a commutative unit ring R in an extension ring S is the set of all elements of S which are integral over R. It is a subring of S containing R.

The subset {0} of a ring. It trivially fulfils the definition of ideal since it is a group (specifically, the zero group), and it is closed under multiplication by any ...