The augmentation map $\epsilon :R[G]\to R$ and the augmentation ideal $I_G=\ker \epsilon$

Let $R$ be a commutative ring and let $G$ be a group. For an element of $R[G]$ written uniquely as a finite sum $x={\sum }_{g\in F}{r}_g[g]$ (The group ring $R[G]$ of finitely supported formal $R$-linear combinations of group elements), define the augmentation map $\epsilon :R[G]⟶R,\epsilon (x):={\sum }_{g\in F}{r}_g\mathrm{.}$

Because the expansion in the basis $\{[g]\}$ is unique, this is a well-defined $R$-linear map. It is also a ring homomorphism in the sense of Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$: additivity is immediate, it sends $[e]$ to $1_R$, and if $x={\sum }_g{r}_g[g]$ and $y={\sum }_h{s}_h[h]$, then $\epsilon (xy)={\sum }_{g,h}{r}_g{s}_h=\left({\sum }_g{r}_g\right)\left({\sum }_h{s}_h\right)=\epsilon (x)\epsilon (y)$ by the multiplication formula supplied by 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 augmentation ideal of $R[G]$ is the kernel $I_G:=\ker \epsilon ,$ which is a two-sided ideal by The kernel of a ring homomorphism is a two-sided ideal.

• The basis element $[g]$ always satisfies $\epsilon ([g])=1_R$.

• When $G$ is finite, the element ${\sum }_{g\in G}[g]$ is the image under the basis sum of the constant function $1$ on $G$.