The characterization of almost prime submodule on the finitely generated module over principal ideal domain (Q6562282)

Let \(R\) be a commutative ring and \(M\) an \(R\)-module. For any submodule \(N\) of \(M\) let \((N:M)=\{r\in R\;|\;rM\subseteq N\}\). A proper submodule \(N\) of \(M\) is called \textit{prime} if, whenever \(r\in R\) and \(m\in M\) such that \(rm\in N\), then either \(m\in N\) or \(r\in (N:M)\). A proper submodule \(N\) of \(M\) is called \textit{almost prime} if, whenever \(r\in R\) and \(m\in M\) such that \(rm\in N-(N:M)N\), then either \(m\in N\) or \(r\in (N:M)\). In the paper under the review the authors give a new approach to the characterization of an almost prime submodule of a finitely generated module over a principal ideal domain. The main result of the paper is that the almost prime submodule must be the direct sum of submodules of the free part and the torsion part of the module, under some conditions.