On co-Hopficity of Injective Envelopes

Abstract

A right R-module M is called co-Hopfian if injective endomorphisms of M R are surjective. It is shown that E(M R ) is co-Hopfian if and only if M R does not contain an infinite direct sum $${{\oplus_{i \in \mathbb{N}}W_{i}}}$$ of submodules such that each W i+1 essentially embeds in W i . For many modules M R , including modules over a right FBN or right duo ring with Krull dimension, it is proved that E(M R ) is co-Hopfian if and only if $${(\mathbb{N})}$$ ↪ M R for every non-zero X R . For a ring which has enough uniforms, the class of modules with co-Hopfian injective envelope is the same as the class of modules with finite uniform dimension if and only if there are only finitely many isomorphism classes of indecomposable injective modules.