The group ring $R[G]$ of finitely supported formal $R$-linear combinations of group elements

Let $R$ be a commutative ring (Commutative ring) and let $G$ be a group (Group and abelian group). The group ring $R[G]$ is the free left $R$-module on the set $G$ (The free module on a set and its standard basis).

For each $g\in G$, write $[g]\in R[G]$ for the standard basis vector indexed by $g$. Thus every element of $R[G]$ has a unique expression ${\sum }_{g\in F}{r}_g[g]$ with $F\subseteq G$ finite and $r_g\in R$.

The notation is deliberately that of formal finite sums: at this stage $R[G]$ is only the underlying free $R$-module with its distinguished basis. The multiplication satisfying $[g][h]=[gh]$ and the resulting $R$-algebra structure are constructed in The group ring $R[G]$ is a unital $R$-algebra with basis $G$, and each $g\in G$ is a unit of $R[G]$.

• The support is finite by definition because $R[G]$ is the direct sum, not the full product, of copies of $R$ indexed by $G$.

• The basis vectors are written $[g]$ to keep the group element $g$ distinct from its basis image in the free module.