Augmentation ideal

In mathematics, an augmentation ideal is an ideal (ring theory) in any group ring. If G is a group and R a commutative ring, there is a ring homomorphism ε from the group ring

R''G''

to R, defined by taking a sum

Σ r_ig_i

to

Σ r_i.

Here r_i is an element of R and g_i an element of G. The sums are finite, by definition of the group ring. In less formal terms,

ε(g)

is defined as 1_R whatever the element g in G, and ε is then extended to a homomorphism of R-modules in the obvious way.

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