Ding modules and dimensions over formal triangular matrix rings

Abstract

Let $T=\bigl(\begin{smallmatrix}A&0\U\&B\end{smallmatrix}\bigr)$ be a formal triangular matrix ring, where $A$ and $B$ are rings and $U$ is a $(B, A)$-bimodule. We prove: (1) If $U\_{A}$ and ${B}U$ have finite flat dimensions, then a left $T$-module $\bigl(\begin{smallmatrix}M\_1\ M\_2\end{smallmatrix}\bigr){\varphi^{M}}$ is Ding projective if and only if $M\_1$ and $M\_2/{\operatorname{im}(\varphi^{M})}$ 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 $\operatorname{FP}$-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.