On semiprimary rings of finite global dimension

Suppose that $A$ is a semiprimary ring satisfying one of the two conditions:

1. its Yoneda ring is generated in finite degrees; 2) its Loewy length is less or equal than three. We prove that the global dimension of $A$ is finite if, and only if, there is a $m>0$ such that $Ex{t}_A^n(S,S)=0$, for all simple $A$-modules $S$ and all $n\ge m$.