On distinguished prime submodules (Q2706524)

The first section of this two-part paper expands on some results by \textit{Abu-Saymeh Sadi} in 1995 regarding properties of \(\mathfrak P\)-prime and primary submodules [Commun. Algebra 23, No. 3, 1131-1144 (1995; Zbl 0833.13004)]. The second part examines conditions for when the classical Krull dimension of an \(R\)-module \(M\) agrees with a dimension defined by Abu-Saymeh S. in the same 1995 paper. NEWLINENEWLINENEWLINEFor specifics, we first let \(M\) be a module of \(R\), a commutative ring with identity, and let \(\mathfrak P\) be a prime ideal of \(R\). Set \(S_{\mathfrak P}=R- {\mathfrak P}\), \({\mathfrak P}M(S_{\mathfrak P}) = \{x\in M : sx\in {\mathfrak P}M \text{ for some }s\in S_{\mathfrak P}\}\) and call a prime submodule \(P\) of \(M\) a \({\mathfrak P}\)-prime submodule if \({\mathfrak P}=(P:M):=\{ r\in R: rM\subseteq P\}\).NEWLINENEWLINENEWLINEPart one of the paper gives conditions on \(M\) which will guarantee that \({\mathfrak P}M(S_{\mathfrak P})\) is a \(\mathfrak P\)-prime submodule of \(M\) and is equal to the intersection of all \(\mathfrak P\)-prime submodules \(N\) for which \({\mathfrak P}= (N:M)\). A result of a similar flavor is also obtained for \(\mathfrak P\)-primary submodules. NEWLINENEWLINENEWLINEIn the second part of this work, the author recalls a definition of dimension for an \(R\)-module \(M\) given by S. Abu-Saymeh in terms of lengths of chains of distinguished prime submodules and shows that this dimension agrees with the classical Krull dimension of \(M\) if \(M\) satisfies either of NEWLINENEWLINENEWLINE(1) \(M\) is a finitely generated distributive module; or NEWLINENEWLINENEWLINE(2) \(M\) is a distributive module with \({\mathfrak M}M\not= M\) over a local ring \((R, {\mathfrak M})\). NEWLINENEWLINENEWLINEA \(\mathfrak P\)-prime submodule \(N\) of \(M\) is called ``distinguished'' if \(N={\mathfrak P}M(S_{\mathfrak P})\).