Gorenstein homological properties of n-trivial extensions of rings

Abstract

We explicitly investigate Gorenstein projective, injective and flat modules over the $n$-trivial extension $R{⋉}_{n}M$ of a ring $R$ by an $R$-bimodule $M$. Assume that $fd(M_R^{{\otimes }_{R}i})<\infty$ and $pd{(}_R{M}^{{\otimes }_{R}i})<\infty$ for any $1\le i\le n$, $fd(\text{Z}(R{)}_{R{⋉}_{n}M})<\infty$ and $pd{(}_{R{⋉}_{n}M}\text{Z}(R))<\infty$. It is proven that a left $R{⋉}_{n}M$-module $(X,f)$ is Gorenstein projective if and only if the sequence $M^{{\otimes }_{R}n+1}{\otimes }_{R}X\stackrel{(M\otimes f)\cdots (M^{{\otimes }_{R}n}\otimes f)}{⟶}M{\otimes }_{R}X\stackrel{f}{⟶}X$ is exact and coker$(f)$ is a Gorenstein projective left $R$-module. As a consequence, we characterize Gorenstein projective, injective and flat modules over tensor rings.