On endomorphism algebras with small homological dimensions

Abstract

We investigate the endomorphism algebras $\Gamma$ of finite dimensional modules having the property that every indecomposable finite dimensional $\Gamma$-module is of projective dimension at most one or injective dimension at most one. In particular, we describe all matrix algebras $\left[\begin{array}{cc}A& 0\\ A& A\end{array}\right]$ with this homological property.