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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.

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    I think that it might be obtained by tensoring from the sequence $R[\mathbb{Z}]\to R[\mathbb{Z}]\to R$ defined by $x\to (s-1)x$, $y\to \epsilon(y)$, where $\epsilon$ is augmentation map.2011-06-03
  • 0
    It looks right to me, but I'm not an expert in homological algebra.2011-06-03

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