Examples of these objects satisfying the descending chain condition are known as Artinian ones.
Two of these objects are said to be Morita-equivalent if there is an equivalence between their module categories. Every local example of one of these objects has a finite Krull dimension and modules are defined over these objects. Examples of these objects satisfying the descending chain condition are known as Artinian ones. The ascending chain condition, the requirement that all ideals be finitely generated and that every ideal contain a maximal element are criteria that are satisfied by Noetherian ones of these. . Commutative examples of these objects have no zero divisors and are called integral domains, of which fields are a subset. For 10 points, identify these algebraic structures exemplified by the integers which unlike groups consist of a set and two binary operations, usually addition and multiplication.
Two of these objects are said to be Morita-equivalent if there is an equivalence between their module categories.
rings