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
, defined by taking a (finite[Note 1]) sum
In less formal terms,
is then extended to a homomorphism of R-modules in the obvious way.
The augmentation ideal A is the kernel of
and is therefore a two-sided ideal in R[G].
A is generated by the differences
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.