Gorenstein homological modules over tensor rings

Abstract

For a tensor ring $T_R(M)$, under certain conditions, we characterize the Gorenstein projective modules over $T_R(M)$, and prove that a $T_R(M)$-module $(X,u)$ is Gorenstein projective if and only if $u$ is monomorphic and ${\rm coker}(u)$ is a Gorenstein projective $R$-module. Gorenstein injective (resp., flat) modules over $T_R(M)$ are also explicitly described. Moreover, we give a characterization for the coherence of $T_R(M)$. Some applications to trivial ring extensions and Morita context rings are given.