\(\psi\)-second submodules of a module
Let \(R\) be a commutative ring with identity and let \(M\) be an \(R\)-module. A non-zero submodule \(N\) of \(M\) is said to be second if for each \(a\in R\), the homomorphism \(\varphi: N\rightarrow N\) with \(\varphi(x)=ax\) for \(x\in N\) is either surjective or zero. In the paper under review the authors introduce and study \(\psi\)-second submodule as follows: Let \(\psi: S(M)\rightarrow S(M)\cup\{\varnothing\}\) be a function, where \(S(M)\) denote the set of all submodules of M. A non-zero submodule \(N\) of \(M\) is called \(\psi\)-second submodule of \(M\) if for any \(r\in R\) and \(K\) a submodule of \(M\), \(rN\subseteq K\), and \(r\psi(N)\nsubseteq K\) implies \(N\subseteq K\) or \(rN = 0\).
scientific article; zbMATH DE number 7227037