Ding modules and dimensions over formal triangular matrix rings

Abstract

Let $T=\biggl(\begin{matrix} A&0\\ U&B \end{matrix}\biggr)$ be a formal triangular matrix ring, where $A$ and $B$ are rings and $U$ is a $(B, A)$-bimodule. We prove that: (1) If $U_A$ and $_B U$ have finite flat dimensions, then a left $T$-module $\biggl(\begin{matrix} M_1\\ M_2\end{matrix}\biggr)_{\varphi^M}$ is Ding projective if and only if $M_1$ and $M_2/{\rm im}(\varphi^M)$ are Ding projective and the morphism $\varphi^M$ is a monomorphism. (2) If $T$ is a right coherent ring, $_{B}U$ has finite flat dimension, $U_{A}$ is finitely presented and has finite projective or $FP$-injective dimension, then a right $T$-module $(W_{1}, W_{2})_{\varphi_{W}}$ is Ding injective if and only if $W_{1}$ and $\ker(\widetilde{\varphi_{W}})$ are Ding injective and the morphism $\widetilde{\varphi_{W}}$ is an epimorphism. As a consequence, we describe Ding projective and Ding injective dimensions of a $T$-module.