Gorenstein homological properties of n-trivial extensions of rings

Abstract

We explicitly investigate Gorenstein projective, injective and flat modules over the $n$-trivial extension $R\ltimes_{n}M$ of a ring $R$ by an $R$-bimodule $M$. Assume that $fd(M^{\otimes_{R}i}_{R})<\infty$ and $pd(_{R}M^{\otimes_{R}i})<\infty$ for any $1\leq i\leq n$, $fd(\textbf{Z}(R)_{R\ltimes_{n}M})<\infty$ and $pd(_{R\ltimes_{n}M}\textbf{Z}(R))<\infty$. It is proven that a left $R\ltimes_{n}M$-module $(X,f)$ is Gorenstein projective if and only if the sequence $M^{\otimes_{R}n+1}\otimes_{R} X\stackrel{(M\otimes f) \cdots(M^{\otimes_{R}n}\otimes f)}\longrightarrow M\otimes_{R} X\stackrel{f}\longrightarrow X$ 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.