On Schur’s Lemma, simple rings and simple-endo modules

Historically speaking, the study of modules whose endomorphism rings are simple (we call them simple-endo modules), has a long history, dating back at least to Schur’s Lemma. This paper attempts to provide a comprehensive study of these modules. Among other things, we complete the characterization of simple-endo modules over commutative rings. We observe that every cyclic [Formula: see text]-module is simple-endo if and only if [Formula: see text] is a simple ring and every proper factor of [Formula: see text] is homogeneous semisimple. Nevertheless, we observe that for a ring [Formula: see text], finitely generated left modules are simple-endo if and only if [Formula: see text] is simple Artinian. Dually, we show that [Formula: see text] is a left V-ring with a unique simple module up to isomorphism if and only if every finitely cogenerated left module is simple-endo. Rings over which every proper cyclic (proper factor) module is simple-endo have also been characterized as well.