Augmentation ideal

In algebra, an augmentation ideal is an ideal that can be defined in any group ring.

If G is a group and R a commutative ring, there is a ring homomorphism $$\varepsilon$$, called the augmentation map, from the group ring $$R[G]$$ to $$R$$, defined by taking a (finite ) sum $$\sum r_i g_i$$ to $$\sum r_i.$$ (Here $$r_i\in R$$ and $$g_i\in G$$.)  In less formal terms, $$\varepsilon(g)=1_R$$ for any element $$g\in G$$, $$\varepsilon(rg)=r$$ for any elements $$r\in R$$ and $$g\in G$$, and $$\varepsilon$$ is then extended to a homomorphism of R-modules in the obvious way.

The augmentation ideal $A$ is the kernel of $$\varepsilon$$ and is therefore a two-sided ideal in R[G].

$A$ is generated by the differences $$ g - g'$$ of group elements. Equivalently, it is also generated by $$\{g - 1 : g\in G\}$$, which is a basis as a free R-module.

For R and G as above, the group ring R[G] is an example of an augmented R-algebra. Such an algebra comes equipped with a ring homomorphism to R. The kernel of this homomorphism is the augmentation ideal of the algebra.

The augmentation ideal plays a basic role in group cohomology, amongst other applications.

Examples of quotients by the augmentation ideal

 * Let G a group and $$\mathbb{Z}[G]$$ the group ring over the integers. Let I denote the augmentation ideal of $$\mathbb{Z}[G]$$. Then the quotient $R[G]$ is isomorphic to the abelianization of G, defined as the quotient of G by its commutator subgroup.
 * A complex representation V of a group G is a $$\mathbb{C}[G]$$ - module. The coinvariants of V can then be described as the quotient of V by IV, where I is the augmentation ideal in $$\mathbb{C}[G]$$.
 * Another class of examples of augmentation ideal can be the kernel of the counit $$\varepsilon$$ of any Hopf algebra.