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

Let R be a commutative Noetherian ring and M a finitely generated R-module. Under various hypotheses, it is proved that the center of EndR(M) coincides with the endomorphism ring of the trace ideal of M. These results are exploited to establish results for balanced and rigid modules, and to settle certain cases of a conjecture of Huneke and Wiegand.