Let $R$ and be a ring with unity, $k$ is a field, and $G$ be a finite group.

Why do we care about ring?

Even we are studying group representation, why do we care about ring?

That is because there is one-to-one correspondence between linear representation of group $G$ in vector space of $V$ over $k$, and **representation space $V$ of $G$.

More generally, we can extend vector space $V$ to $R$G-module $V$, and $G{L}_n(V)$ to $Au{t}_R(V)$.

Also we can use ring theory to Group Representation.