Augmentation ideal

In mathematics, an augmentation ideal is an ideal 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 is the kernel of ε, and is therefore a two-sided ideal in R[G]. It is generated by the differences

g − g′

of group elements.

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.

Another class of examples of augmentation ideal can be the kernel of the counit ε of any Hopf algebra.

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