Retractable and coretractable modules in Wisbauer category

Abstract

A right R-module M is called: (1) retractable if HomR(M,N)≠0 for any non-zero submodule N of M; (2) coretractable if HomR(M/N,M)≠0 for any proper submodule N of M. It shows that if M is locally noetherian and every nonzero module in the category σ[M] has a maximal submodule, then the retractability and coretractability of modules in σ[M] coincide. Let C be a coalgebra over a field k. We prove that all right C-comodules are retractable if and only if every right C-comodule is coretractable.