That type is termed Noetherian.
The Dedekind example of this is commutative and is equivalent to a set of algebraic integers in its field of fractions. Schur's lemma states that a specific type of these for an irreducible module is a division algebra. Another example of this structure follows the equivalent statements that certain subsets of it each contain a maximal element and follow the ascending chain condition. That type is termed Noetherian. Commutative examples of this structure with no zero divisors are termed integral domains. All modules are defined over this structure, and subsets of this structure that form abelian groups under addition and absorb products are termed ideals. For 10 points, name this algebraic structure consisting of a set with two operations, usually addition and multiplication, where the set forms an additive abelian group.
The Dedekind example of this is commutative and is equivalent to a set of algebraic integers in its field of fractions.
{ring}