When is tight closure determined by the test ideal?

TL;DR

This paper characterizes rings where tight closure equality involving the test ideal holds universally, showing under certain conditions they are either weakly F-regular or one-dimensional.

Contribution

It provides a characterization of rings based on tight closure and test ideal conditions, identifying specific structural properties.

Findings

Rings satisfying the tight closure equality are either weakly F-regular or one-dimensional under certain assumptions.

The paper offers a criterion to determine when tight closure is governed by the test ideal.

It advances understanding of the relationship between tight closure, test ideals, and ring regularity.

Abstract

We characterize the rings in which the equality $(\tau I:\tau)= I^*$ holds for every ideal $I \subset R$. Under certain assumptions, these rings must be either weakly F-regular or one-dimensional.