Topics on sequentially Cohen-Macaulay modules (Q1657985)

Let \(R\) be a commutative Noetherian ring. In the paper under review, the authors investigates the following questions: (1) whether the sequentially Cohen-Macaulay property is inherited from localizations; (2) whether the sequentially Cohen-Macaulay property preserves the module-finite extension of rings. In this regards they succeed to prove the following results: Theorem 1.1. Suppose that \(\dim R/\mathfrak{p}=\dim R_{\mathfrak{m}}/\mathfrak{p}R_{\mathfrak{m}}\) for every \(\mathfrak{p}\in \mathrm{Ass}_R(M)\) and for every maximal ideal \(\mathfrak{m}\) of \(R\) such that \(\mathfrak{p}\subseteq\mathfrak{m}\). Then the following conditions are equivalent: {\parindent=6mm\begin{itemize}\item[(1)] \(M\) is a sequentially Cohen-Macaulay \(R\)-module. \item[(2)] \(M_p\) is a sequentially Cohen-Macaulay \(R_p\)-module for every \(p\in \mathrm{Supp}_R(M)\). \end{itemize}} Theorem 1.2. Suppose \(R\) is a local ring. Let \(R\ltimes M\) denote the idialization of \(M\) over \(R\). Then the following conditions are equivalent: {\parindent=6mm \begin{itemize}\item[(1)] \(R\ltimes M\) is sequentially Cohen-Macaulay local ring. \item[(2)] \(R\ltimes M\) is sequentially Cohen-Macaulay \(R\)-module. \item[(3)] \(R\) is a sequentially Cohen-Macaulay local ring, and \(M\) is a sequentially Cohen-Macaulay \(R\)-module. \end{itemize}}