I have a reasonably precise question which I hope is clear enough to get a nice answer. Let R be a Noetherian non-commutative ring which is finite as a module (and flat/free if it helps) over it's center Z(R) which we can assume has finite Krull dimension. One can also assume R integral over Z(R). By a two sided prime ideal, I mean a two sided ideal P where if $xRy$ is contained in P, then either x and/or y are in P. Consider now the abelian category of left modules and suppose we know that $Ext^m(R/P,M)=0$ for any M and $m>n$ for some n (that is independent of P). Now there aren't enough (two-sided) prime ideals in a general non-commutative ring, but there are a fair number in rings such as the one I describe above.

**Question: Does it follow that for any finitely generated left module N, $Ext^m(N,M)=0$ for $m>n'$ which can depend on N **. The issue I am having is that we don't have quite as effective a filtration sequence of any N, we only have a filtration such that $N_i/N_{i-1}=I/P$ for some left ideal and a prime P. That might be the end of the story, but there might be other tricks I am not aware of. Either way, I haven't been able to sort it out or find a good reference. If there is an extra hypothesis that helps the situation, I'd like to know about it.