On modules satisfying \(S\)-dccr condition
Let \(A\) be a commutative ring with \(0 \neq 1\) and \(X\) a unital \(A\)-module. Suppose that \(S \subseteq A\) is amultiplicatively closed subset. \(X\) is said to satisfy \(S\)-dccr (\(S\)-dccr\(^{\star}\)) condition if for each finitely generated (principal) ideal \(I\) of \(A\) and a submodule \(Y\) of \(X\), the descending chain \(\{I^iY\}_{i \in \mathbb{N}}\) is \(S\)-stationary. In this paper, the authors investigated the behaviour of S-dccr (\(S\)-dccr\(^{\star}\))-condition under localization, homomorphism, factor module and in cartesian product of modules (See Lemma 1, Propositions 10, 11 and Theorem 13). Also, they proved that the conditions \(S\)-dccr and \(S\)-dccr\(^{\star}\) are equivalent for every \(A\)-module \(X\) (see Theorem 4). In last section, The \(S\)-versions of the Union Theorem and Nakayama's Lemma were discussed in this section in the context of the \(S\)-dccr notion (see Proposition 16 and Theorem 17).
scientific article; zbMATH DE number 7608930