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 formula_1, called the augmentation map, from the group ring formula_2 to formula_3, defined by taking a (finite) sum formula_4 to formula_5 (Here formula_6 and formula_7.) In less formal terms, formula_8 for any element formula_9, formula_10 for any element formula_11, and formula_1 is then extended to a homomorphism of "R"-modules in the obvious way. 
The augmentation ideal is the kernel of formula_1 and is therefore a two-sided ideal in "R"["G"]. 
 is generated by the differences formula_14 of group elements. Equivalently, it is also generated by formula_15, 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.