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

The augmentation ideal A is the kernel of ε 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 ∈ 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.