The usual group cohomology $H^n(G, M) = \text{Ext}_{\mathbb{Z}[G]}^n(\mathbb{Z}, M)$ and can be computed via canonical chain complex $$\cdots \rightarrow \mathbb{Z}[G^{n+1}] \rightarrow \cdots \rightarrow \mathbb{Z} \rightarrow 0$$
Is there similar cohomology theory where we replace $\mathbb{Z}$ by a different ring such as $\mathbb{Z}_p$?