Trace Ideals and Centers of Endomorphism Rings of Modules over Commutative Rings

TL;DR

This paper investigates the relationship between the center of endomorphism rings and trace ideals of modules over commutative Noetherian rings, providing new insights into module theory and related conjectures.

Contribution

It proves that under various conditions, the center of the endomorphism ring equals the endomorphism ring of the trace ideal, advancing understanding of module endomorphisms.

Findings

Center of End_R(M) equals End_R(trace of M) under certain conditions

Results applied to balanced and rigid modules

Partial resolution of a conjecture by Huneke and Wiegand

Abstract

Let $R$ be a commutative Noetherian ring and $M$ a finitely generated $R$-module. Under various hypotheses, it is proved that the center of $\mbox{End}_R(M)$ 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.