A group ring $R[G]$ is a ring constructed from a group $G$ and ring $R$. A special case of this construction is group algebra, which occurs naturally in representation theory.

The group ring $R[G]$ is constructed in the following way. The set $R[G]$ is the free $R$-module on the elements of $G$, equipped with the multiplication given by the operation in $G$ extended distributively to all elements in the free module.

A special case of this construction is group algebra, which occurs naturally in representation theory. It turns out that every group representation $\rho:G\rightarrow GL(V)$ corresponds to an $R[G]$ module structure on $V$. This connection ties the representation theory of groups to the module theory of group algebras.

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