$\Omega$-Gorenstein Modules over Formal Triangular Matrix Rings

Abstract

Let A and B be rings and U a (B, A)-bimodule. Under some conditions, $\Omega$-Gorenstein module over the formal triangular matrix ring $T=\left( \begin{array}{cc} A \,\ &{}\quad 0 \\ U \ &{}\quad B \\ \end{array} \right)$ is explicitly described, where $\Omega$ is a class of left T-modules. As an application, it is shown that if $_BU$ has finite projective dimension and $U_A$ has finite flat dimension, then $M=\left( \begin{array}{c} M_1 \\ M_2 \\ \end{array} \right) _{\varphi ^M}$ is a Gorenstein projective left T-module if and only if $M_1$ is a Gorenstein projective left A-module,${{\text {Coker}}(}\varphi ^M)$ is a Gorenstein projective left B-module and $\varphi ^M:{U\otimes _{A}M_1}\rightarrow M_2$ is a monomorphism. This statement covers an earlier result of Enochs, Cortés-Izurdiaga and Torrecillas in this direction.