Regularity, local cohomology and injective modules under contracting endomorphisms

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