Correct classes of modules

Robert Wisbauer

Algebra and discrete mathematics, no. 4 (2004), pp. 106-118.

Résumé

For a ring $R$, call a class $\mathcal{C}$ of $R$-modules (pure-) mono-correct if for any $M,N\in\mathcal {C}$ 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 $R$-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.

Détail

Keywords: Cantor-Bernstein Theorem, correct classes, homological classification of rings.