On the converse of Schur's Lemma for modules over their endomorphism rings

We investigate right modules over commutative rings whose biendomorphism rings are division rings. First, we provide a complete characterization of such modules. We then establish a Schur-type converse in this setting, showing a precise interplay between a module and its biendomorphism ring. Specifically, let R be a commutative ring, M a right R-module, and L = End R (M). We prove that M, regarded as a left L-module, is simple if and only if End L (M) = BiEnd R (M) is a division ring. This result highlights that, for modules over commutative rings, simplicity over the endomorphism ring is equivalent to having a division biendomorphism ring, providing a clear module-theoretic converse to Schur's lemma.