Homological aspects of formal triangular matrix rings

Authors : Lixin MAO

Pages : 1504-1516

Doi:10.15672/hujms.1014028

View : 23 | Download : 7

Publication Date : 2022-12-01

Article Type : Research Paper

Abstract :Let $T=\bigglinsert ignore into journalissuearticles values(\begin{matrix} A&0\\U&B\end{matrix}\biggr);$ be a formal triangular matrix ring, where $A$ and $B$ are rings and $U$ is a $insert ignore into journalissuearticles values(B, A);$-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 insert ignore into journalissuearticles values(weak); global dimensions of $T$. It is proven that insert ignore into journalissuearticles values(1); If $U_{A}$ is flat and $_{B}U$ is projective, $lDinsert ignore into journalissuearticles values(A);\neq lDinsert ignore into journalissuearticles values(B);$, then $lDinsert ignore into journalissuearticles values(T);={\rm max}\{lDinsert ignore into journalissuearticles values(A);,lDinsert ignore into journalissuearticles values(B);\}$; insert ignore into journalissuearticles values(2); If $U_{A}$ and $_{B}U$ are flat, $wDinsert ignore into journalissuearticles values(A);\neq wDinsert ignore into journalissuearticles values(B);$, then $wDinsert ignore into journalissuearticles values(T);={\rm max}\{wDinsert ignore into journalissuearticles values(A);,wDinsert ignore into journalissuearticles values(B);\}$.
Keywords : formal triangular matrix ring, projective dimension, injective dimension, flat dimension, FP injective dimension