Epis and monos which must be isos

Abstract

Orzech [1] has shown that every surjective endomorphism of a noetherian module is an isomorphism. Here we prove analogous results for injective endomorphisms of noetherian injective modules, and the duals of these results. We prove that every injective endomorphism, with large image, of a module with the descending chain condition on large submodules is an isomorphism, which dualizes a result of Varadarajan [2]. Finally we prove the following result and its dual: if p is any radical then every surjective endomorphism of a module M, with kernel contained in pM, is an isomorphism, provided that every surjective endomorphism of pM is an isomorphism.