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 to \sum r_i. (Here r_i\in R and g_i\in G.) In less formal terms, for any element g\in G, 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, 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

I/I2 is isomorphic to the abelianization of G, defined as the quotient of G by its commutator subgroup.