Regularity, local cohomology and injective modules under contracting endomorphisms

Kamran Divaani-Aazar; Mohammad Eghbali

commutative algebra

Regularity, local cohomology and injective modules under contracting endomorphisms

arXiv:2607.13320

summary

The paper generalizes results known for rings of characteristic p to any ring that admits a contracting endomorphism, using Hom and tensor functors to give new characterizations of regular rings and to analyze the effects on local cohomology and injective modules.

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 $I ¨ : R \to R$ be such an endomorphism, and denote by $^{I ¨} R$ the ring $R$ endowed with the left $R$ -module structure induced by $I ¨ $ . We employ the functor $Hom_{R} (^{I ¨} R, -)$ to derive new characterizations of regular rings. Furthermore, we examine the behavior of the functor $(-) \otimes_{R}^{I ¨} R$ on local cohomology modules and injective modules.

17 pages

Topics & keywords

#contracting endomorphisms #regular rings #local cohomology #injective modules #module functorscontracting endomorphismHom_R(^φR,-)tensor_R(^φR)regular ring characterizationlocal cohomology modulesinjective module behavior