Ding modules and dimensions over formal triangular matrix rings

Abstract

Let $T=\left(\begin{array}{cc}A& 0\\ U& B\end{array}\right)$ 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 $(\begin{array}{c}{M}_1\\ M_2\end{array}{)}_{{\phi }^M}$ is Ding projective if and only if $M_1$ and $M_2/\mathrm{im}({\phi }^M)$ are Ding projective and the morphism ${\phi }^M$ 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 $(W_1,W_2{)}_{{\phi }_W}$ is Ding injective if and only if $W_1$ and $\ker (\widetilde{{\phi }_W})$ are Ding injective and the morphism $\widetilde{{\phi }_W}$ is an epimorphism. As a consequence, we describe Ding projective and Ding injective dimensions of a $T$-module.