Test ideal restricted to principal ideals and connections to trace ideals

TL;DR

This paper explores the relationship between test ideals restricted to principal ideals and trace ideals, establishing that such test ideals are always trace ideals and linking them to the trace of the center of endomorphism rings.

Contribution

It demonstrates that test ideals restricted to principal ideals are trace ideals and connects them to the trace of the center of endomorphism rings for torsionless, faithful modules.

Findings

Test ideals restricted to principal ideals are trace ideals.

When modules are torsionless and faithful, the test ideal equals the trace of the center of their endomorphism ring.

Provides applications illustrating these theoretical results.

Abstract

Motivated by some recent results of F. P\'erez and R. R.G connecting test ideal of module closure operations and trace ideals, we investigate the test ideal restricted to principal ideals corresponding to a module closure operation of a torsion-free module $M$. We show that such a test ideal is always a trace ideal, and moreover when $M$ is torsionless and faithful, then it is the trace of the center of $\operatorname{End}_R(M)$. Some applications are given.