Regularity, local cohomology and injective modules under contracting endomorphisms

Abstract

This work extends several fundamental results previously established for rings of characteristic $p$ to the broader class of rings that admit a contracting endomorphism. Specifically, let $\varphi :R\to R$ be such an endomorphism, and denote by ${}^{\varphi }R$ the ring $R$ endowed with the left $R$-module structure induced by $\varphi$. We employ the functor ${\text{Hom}}_R{(}^{\varphi }R,-)$ to derive new characterizations of regular rings. Furthermore, we examine the behavior of the functor $(-){\otimes }_R{}^{\varphi }R$ on local cohomology modules and injective modules.

17 pages