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, called the augmentation map, from the group ring to, defined by taking a sum to In less formal terms, for any element, for any element, and is then extended to a homomorphism of R-modules in the obvious way.
The augmentation ideal is the kernel of and is therefore a two-sided ideal in R.
is generated by the differences 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 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