- Hacettepe Journal of Mathematics and Statistics
- Vol: 51 Issue: 6
- Homological aspects of formal triangular matrix rings
Homological aspects of formal triangular matrix rings
Authors : Lixin Mao
Pages : 1504-1516
Doi:10.15672/hujms.1014028
View : 19 | Download : 6
Publication Date : 2022-12-01
Article Type : Research
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 first give some computing formulas of projective, injective, flat and $FP$-injective dimensions of special left $T$-modules. Then we establish some formulas of (weak) global dimensions of $T$. It is proven that (1) If $U_{A}$ is flat and $_{B}U$ is projective, $lD(A)\neq lD(B)$, then $lD(T)={\rm max}\{lD(A),lD(B)\}$; (2) If $U_{A}$ and $_{B}U$ are flat, $wD(A)\neq wD(B)$, then $wD(T)={\rm max}\{wD(A),wD(B)\}$.Keywords : formal triangular matrix ring, projective dimension, injective dimension, flat dimension, FP-injective dimension