Semidualizing modules and trivial ring extensions

Abstract

Let $R \ltimes M$ be a trivial extension of a commutative ring $R$ by an $R$-$R$-bimodule $M$. We first investigate some homological properties over $R \ltimes M$. Then we provide a way to construct new semidualizing $R \ltimes M$-modules from given semidualizing $R$-modules. It is proven that $C$ is a semidualizing $R$-module and $M$ belongs to the Auslander class $\mathcal{A}_{C}(R)$ if and only if $\mathbf{T}(C)$ is a semidualizing $R \ltimes M$-module and $\textrm{Tor}_{i}^{R}(C,M) = 0$ for any $i \geq 1$. Finally, we describe when $\mathbf{T}(C)$ is a dualizing $R \ltimes M$-module.