# Ding modules and dimensions over formal triangular matrix rings

### Lixin Mao

Nanjing Institute of Technology, China

## Abstract

Let $T=(AU 0B )$ 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 $(M_{1}M_{2} )_{φ_{M}}$ 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.

## Cite this article

Lixin Mao, Ding modules and dimensions over formal triangular matrix rings. Rend. Sem. Mat. Univ. Padova 148 (2022), pp. 1–22

DOI 10.4171/RSMUP/100