Suppose R has finite global dimension n, N is a f.g. module, F is a free module and Ext^n(N,F) is not equal to zero, then Ext^n(N, R) is not trival either.
Note, HERE R is not Noetherian necessarily.
Suppose R has finite global dimension n, N is a f.g. module, F is a free module and Ext^n(N,F) is not equal to zero, then Ext^n(N, R) is not trival either. Note, HERE R is not Noetherian necessarily. 


This cannot be right if $R$ is injective as an $R$module and the global dimension is nonzero but finite. Let for instance $R$ be the ring $R = \prod_{i=1}^\infty \mathbb{C}$ and assume the Continuum Hypothesis so that ${\rm gldim}(R)=2$. Then by a theorem of Auslander there is a cyclic module $N$ such that ${\rm pd}(N)=2$. Take a free resolution of $N$ and obtain a free module $F$ such that ${\rm Ext}^2(N,F) \neq 0$. But ${\rm Ext}^2(N,R)=0$ since $R$ is injective. 

