Correct classes of modules
نویسنده
چکیده
For a ring R, call a class C of R-modules (pure-) mono-correct if for any M,N ∈ C the existence of (pure) monomorphisms M → N and N → M implies M ' N . Extending results and ideas of Rososhek from rings to modules, it is shown that, for an R-module M , the class σ[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 thatR is left pure semisimple if and only ifR-Mod is pure-mono-correct. Furthermore, the class of pure-injective Rmodules 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.
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