Let $R$ be a ring and $M$ an $R$-module then
inj dim $M\leq i\in\mathbb{N}$ if and only if $\mathrm{Ext}^{i+1}(N,M)=0$ for every cyclic module $N$.
The implication from left to right is obvious, I'm finding some difficulties in proving the other implication. In case we need it I think we can suppose $M$ finitely generated and $R$ noetherian. Do you have any suggestions fot the implication I'm missing?