Trace ideals and the centers of endomorphism rings of modules over commutative rings

Let $R$ be a commutative Noetherian ring and $M$ a finitely generated $R$-module. We establish cases in which the centers of $\End R M$ and $\End R {M^*}$ coincide with the endomorphism ring of the trace ideal of $M$. These observations are exploited to prove results for balanced and rigid modules, as well as modules with $R$-free endomorphism rings. As a consequence, we clarify the relationship between the properties of $M$ and those of its endomorphism rin