Modules and rings satisfying strong accr (Q6651071)

Let \(R\) be a commutative ring and \(M\) an \(R\)-module. We say that \(M\) satisfies accr (respectively, satisfies accr\(^\ast\)) if the ascending chain of submodules of the form \(N:_MB\subseteq N:_MB^2\subseteq N:_M B^3\subseteq\cdots\) terminates for every submodule \(N\) of \(M\) and every finitely generated (respectively, principal) ideal \(B\) of \(R\). The class of modules satisfying accr (equivalently, accr\(^\ast\)) has been extensively studied in the literature and is known to be quite large. It contains, among others, Noetherian modules, modules over Artinian rings, modules having ACC on colon submodules, Laskerian modules, and modules over perfect rings. Other variants of modules satisfying accr (respectively, satisfies accr\(^\ast\)) have been also defined, including, among others, strong accr\(^\ast\), \(S\)-accr, and \(S\)-accr\(^\ast\), where \(S\) is a multiplicative set of \(R\). These modules are known to share various important properties with Noetherian modules. \N\NThis article explore more rings and modules that satisfy the property of strong accr\(^\ast\). A module \(M\) satisfies strong accr\(^\ast\) (or (C)) if for any submodule \(N\) of \(M\) and for any sequence \(\langle r_n\rangle\) of elements of \(R\), the ascending sequence of submodules of the form \(N:_M r_1\subseteq N :_M r_1r_2\subseteq N :_M r_1r_2r_3\subseteq\cdots\) is stationary. Among the main results, the necessary and sufficient conditions for various types of rings -- such as valuation rings, trivial extensions, the pullbacks, and amalgamated algebras -- to satisfy strong accr\(^\ast\) are established.