$(m,n)$-Igusa-Todorov Algebras, IT-Dimensions and Triangular Matrix Algebras

Abstract:

      Let $T$, $U$ be two Artin algebras and $_U M_T$ be a $U$-$T$-bimodule. In this paper, we get a necessary and sufficient condition such that the formal triangular matrix algebra $\Lambda=\left({\smallmatrix T&0\\ M&U\endsmallmatrix}\right)$ is $(m,n)$-Igusa-Todorov when $_U M$, $M_T$ are projective. We also study the Igusa-Todorov dimension of $\Lambda$. More specifically, it is proved that $$\max\{\ITdim T, \ITdim U\}\leqslant\ITdim \Lambda\leqslant\min\{\max\{\gldim T,\ITdim U\},\max\{\gldim U,\ITdim T\}\}.$$