Let $R$ be a ring and $R[\mathbb{Z}]$ be the group ring obtained from ring $R$ and group $\mathbb{Z}=$.
Suppose that $M$ be a $R[\mathbb{Z}]$-module and it is isomorphic to $R^n$ as $R$-module.
We can describe $R[\mathbb{Z}]$-module structure as a matrix $H\colon R^n\to R^n$ which corresponds to a map $s\colon M\to M$ defined by $v\to s\cdot v$.
Now, we can regard $H$ a $n\times n$ square matrix over $R[\mathbb{Z}]$ via the inclusion $R\to R[\mathbb{Z}]$.
Is it true that $M\cong \operatorname{Coker}(sI-H\colon R[\mathbb{Z}]^n\to R[\mathbb{Z}]^n)$? (I think that it should be.)
This is not the homework.