General form of Maschke's Theorem for $R[G]$

Question

I've come across some reference to a form of Mashke's theorem for rings; namely, $R[G]$ is semisimple if $R$ is a commutative ring (not necessarily a field) and $G$ is a finite group such that $|G|$ is a unit in $R$. However, I can't seem to find any source for this. Is this true? Where can I find a proof?

Answer 1

This is not true as stated, e.g. because if we take $G$ to be trivial, then you would be stating that an arbitrary commutative ring $R$ is semi-simple, which is false.

But, if you assume that $R$ is commutative semi-simple, and that $|G|$ is a unit in $R$, then $R[G]$ is semi-simple. Since a semi-simple commutative ring is a product of fields, this quickly reduces to the case when $R$ is a field (in which $|G|$ is non-zero), and the usual argument with averaging over the elements of the group (possible because we can divide by $|G|$ in $R$) proves semi-simplicity.