Weakly \(N\)-injective modules
Properties of weak \(N\)-injectivity as a generalisation of \(N\)-injectivity [see \textit{T. Albu} and \textit{C. Nastasescu}, Relative finiteness in module theory (Pure Appl. Math. 84, 1984; Zbl 0556.16001)] are studied. Given two right \(R\)-modules \(M\) and \(N\), we say that \(M\) is weakly \(N\)-injective if for every \(f:N\to E(M)\), there exists a monomorphism \(h:M\to E(M)\) and a homomorphism \(g:N\to M\) such that \(f=hg\). If \(R\) is a PID, let \(R_{p^\infty}\) denote the set of all \((\frac{a}{p^n})+R\), where \(a\in R\), \(n\in\mathbb{Z}\) and \(p\) is a prime in \(R\). A right \(R\)-module \(X\) is found to be weakly \(R_{p^\infty}\)-injective if and only if \(X\) is isomorphic to \((\oplus R_{p^\infty})\oplus Y\), where \(Y\) has no \(p\)-torsion element. Examples show that weak \(N\)-injectivity is indeed a generalization of \(N\)-injectivity. The class of all \(N\)-injective modules, for any given \(N\), is shown to be closed under finite direct sums, essential extensions and arbitrary direct products. Necessary and sufficient conditions are derived for this class to be closed under arbitrary direct sums. On the other hand, the class of all modules \(N\) with respect to which an \(R\)-module \(M\) is weakly \(N\)-injective, is proved to be closed under submodules, quotient modules and arbitrary direct sums.
scientific article; zbMATH DE number 1005380