Hi,

Let $A$ be a ring. it is a well known fact that if $Ext^v_A(M,A)=0$ for every $v>\mu$ and every left module $M$ then $inj.dim(A)\leq \mu$ (seeing $A$ as a left $A$-module).

Does the weaker hypothesis $Ext^v_A(M,A)=0$ for every $v>\mu$ and every **finitely generated** left module $M$ imply the same result, at least when $A$ is Noetherian?

Thank you in advance.