Correct classes of modules

Abstract

For a  ring \(R\), call a class \(\cal C\) of \(R\)-modules  (pure-) mono-correct if for any \(M,N \in \cal C\) the existence of (pure) monomorphisms \(M\to N\) and \(N\to M\) implies \(M\simeq N\). Extending results and ideas of Rososhek from rings to modules, it is shown that, for an \(R\)-module \(M\), the class \(\sigma [M]\) of all \(M\)-subgenerated modules is mono-correct if and only if \(M\) is semisimple, and the class of all weakly \(M\)-injective modules is mono-correct if and only if \(M\) is locally noetherian. Applying this to the functor ring of \(R\)-Mod provides a new proof that \(R\) is left pure semisimple if and only if \(R\mbox{-Mod}\) is pure-mono-correct. Furthermore, the class of pure-injective \(R\)-modules is always pure-mono-correct, and it is mono-correct if and only if \(R\) is von Neumann regular. The dual notion epi-correctness is also considered and it is shown that a ring \(R\) is left perfect if and only if the class of all flat  \(R\)-modules is epi-correct. At the end some open problems are stated.