Ding modules and dimensions over formal triangular matrix rings

Mao, Lixin

Rendiconti del Seminario Matematico della Università di Padova, Tome 148 (2022), pp. 1-22.

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: (1) If U A and B U have finite flat dimensions, then a left T-module ${\left(\begin{array}{c}{M}_1\\ M_2\end{array}\right)}_{{\phi }^M}$ is Ding projective if and only if M 1 and M 2 /im(φ M ) are Ding projective and the morphism φ 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 ) φ W is Ding injective if and only if W 1 and ker(φ W ˜) are Ding injective and the morphism φ W ˜ is an epimorphism. As a consequence, we describe Ding projective and Ding injective dimensions of a T-module.