Injectivity relative to closed submodules.
Let \(R\) be a ring. An \(R\)-module \(X\) is called \(c\)-injective if, for every closed submodule \(L\) of every \(R\)-module \(M\), every homomorphism from \(L\) to \(X\) lifts to \(M\). The following two results are obtained in this paper: (1) If \(R\) is a Dedekind domain, then an \(R\)-module \(X\) is \(c\)-injective iff \(X\) is isomorphic to a direct product of homogeneous semisimple \(R\)-modules and injective \(R\)-modules. (2) A commutative Noetherian domain \(R\) is Dedekind iff every simple \(R\)-module is \(c\)-injective.
scientific article; zbMATH DE number 5545847
scientific article