---
abstract: 'The aim of this paper is to introduce a new class of Noetherian rings of positive characteristic in terms of perfect closures and study their basic properties. If the perfect closure of a Noetherian ring is coherent, we call it an $F$-coherent ring. Some interesting applications are given in connection with tight closure theory. In particular, we discuss relationships between $F$-coherent rings and $F$-pure, $F$-regular, and $F$-injective rings. The final section discusses how the coherent property effects the behavior of tight closure for general perfect rings.'
author:
- Kazuma Shimomoto
title: 'F-coherent rings with applications to tight closure theory'
---

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Introduction
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In the present paper, all rings are assumed to be commutative and unitary. Recall that a ring is $\textit{coherent}$ if every finitely generated ideal is finitely presented. This condition is automatic for all Noetherian rings. We shall consider the following problem in connection with tight closure theory.

Let $R$ be a Noetherian ring of positive characteristic and let $R^{\infty}$ be its perfect closure. Then what is characteristic of $R$ if $R^{\infty}$ is a coherent ring?

Tight closure theory was created in the mid 80’s by Hochster and Huneke as a powerful tool with applications to 