The Frobenius functor and injective modules

TL;DR

This paper studies rings in prime characteristic where the Frobenius functor preserves injective modules, characterizing one-dimensional cases and including all one-dimensional F-pure rings.

Contribution

It provides a characterization of one-dimensional local rings with Frobenius preserving injectivity, expanding understanding of F-pure rings in prime characteristic.

Findings

Characterized one-dimensional local rings with Frobenius preserving injective modules.

Showed all one-dimensional F-pure rings have this property.

Identified classes of rings where Frobenius functor maintains injectivity.

Abstract

We investigate commutative Noetherian rings of prime characteristic such that the Frobenius functor applied to any injective module is again injective. We characterize the class of one-dimensional local rings with this property and show that it includes all one-dimensional F-pure rings.