On \(S\)-strong Mori modules
The Wang and McCasland definition of a strong Mori module (SM-module) [\textit{F. Wang} and \textit{R. L. McCasland}, J. Pure Appl. Algebra 135, No. 2, 155--165 (1999; Zbl 0943.13017)] was generalized by Kim, Kim and Lim [\textit{H. Kim} et al., J. Algebra 416, 314--332 (2014; Zbl 1304.13038)] as follows. Let \(A\), be a domain, \(S\) be a multiplicative subset of \(A\) and \(M\) be a \(w\)-module over \(A\). The module \(M\) is \textit{\(S\)-\(w\)-finite} if \(sM \subseteq F\) for some element \(s\in S\) and some \(w\)-finite type submodule \(F\) of \(M\). Also, \(M\) is an \textit{\(S\)-strong Mori} module (S-SM-module) if each \(w\)-submodule of \(M\) is \(S\)-\(w\)-finite. The domain \(A\) is \textit{\(S\)-strong Mori} if and only if \(A\) is a \(S\)-strong Mori \(A\)-module. The authors show that the domain \(\mathbb Z+X\mathbb Q[X]\) is \(S\)-strong Mori for \(S=R\setminus \{0\}\), but it is not strong Mori. The authors investigate properties related to the \(S\)-strong Mori property. Among other results, they prove the following for a \(w\)-module \(M\): if \(M\) is \(S\)-strong Mori, then every increasing sequence of \(w\)-finite type submodules of \(M\) is \(S\)-stationary, and the converse is true if the set \(S\) is countable. Here, an ncreasing sequence \((N_n ) _{n\in \mathbb N}\) of submodules of \(M\) is called \textit{\(S\)-stationary} if there exist a positive integer \(k\) and \(s\in S\) such that for each \(n\geq k\), we have \(sN_n\subseteq N_k\).
scientific article; zbMATH DE number 6983436