Let $R$ be a ring and $G$ an infinite group. The $R$-cohomological dimension of $G$ can be defined as $cd_R(G) = \min \{ n \geq 0 : H^{n+1}(G, R[G]) = 0\}$, which implies that $H^i(G, M) = 0$ for all $i > cd_R(G)$ and any $R[G]$-module $M$.
Here $R[G]$ denotes the group ring, which as a module is simply the direct sum of copies of $R$ indexed by $G$, with the action of $G$ permuting the coordinates. Denote instead by $R(G)$ the direct product of copies of $R$ indexed by $G$ with the same action (is there a standard notation?). This is not a ring anymore but it is still an $R[G]$-module, and so we can ask when does $H^n(G, R(G)) = 0$.
My question is,can we use this module to compute the cohomological dimension instead? That is, does $cd_R(G) = \min \{ n \geq 0 : H^{n+1}(G, R[G]) = 0\}$? Or what are some examples in which this does not hold? Does it hold under some finiteness hypothesis on the group (say, finitely generated, finitely presented...)
In case it helps I am interested in particular in the case in which $R$ is finite, or even a finite field, and $n = 1$.
$H^n(G,R(G))$ is always zero for $n>0$.
As Qiaochu Yuan pointed out in comments, $R(G)=\operatorname{Hom}_R(R[G],R)$. So $$\operatorname{Hom}_{R[G]}(?, R(G))\!=\operatorname{Hom}_{R[G]}(?,\operatorname{Hom}_R(R[G],R))=\operatorname{Hom}_R(?\otimes_{R[G]}R[G],R)= \operatorname{Hom}_R(?,R).$$
So this shows that, calculating $H^n(G,R(G))=\operatorname{Ext}^n_{R[G]}(R,R(G))$ by applying $\operatorname{Hom}_{R[G]}(?, R(G))$ to a projective $R[G]$-module resolution of $R$, it is isomorphic as an $R$-module to $\operatorname{Ext}^n_R(R,R)$, which is zero for $n>0$.