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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 $$\varepsilon$$ , called the augmentation map, from the group ring R[G] to R, defined by taking a (finite[Note 1]) sum $$\sum r_{i}g_{i}$$ to $$\sum r_{i}$$. (Here $$r_{i}\in R$$ and $${\displaystyle g_{i}\in G}$$ .) In less formal terms,$${\displaystyle \varepsilon (g)=1_{R}}$$ for any element $$g\in G$$ , $${\displaystyle \varepsilon (r)=r}$$ for any element $$r\in R$$, and $$\varepsilon$$ is then extended to a homomorphism of R-modules in the obvious way.

The augmentation ideal A is the kernel of $$\varepsilon$$ and is therefore a two-sided ideal in R[G].

A is generated by the differences g-g' of group elements. Equivalently, it is also generated by $${\displaystyle \{g-1:g\in G\}}$$, 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.
Examples of Quotients by the Augmentation Ideal

Let G a group and $${\displaystyle \mathbb {Z} [G]}$$ the group ring over the integers. Let I denote the augmentation ideal of $${\displaystyle \mathbb {Z} [G]}$$. Then the quotient I/I2 is isomorphic to the abelianization of G, defined as the quotient of G by its commutator subgroup.
A complex representation V of a group G is a $${\displaystyle \mathbb {C} [G]}$$ - module. The coinvariants of V can then be described as the quotient of V by IV, where I is the augmentation ideal in $${\displaystyle \mathbb {C} [G]}$$.
Another class of examples of augmentation ideal can be the kernel of the counit $$\varepsilon$$ of any Hopf algebra.

Notes

When constructing R[G], we restrict R[G] to only finite (formal) sums

References

D. L. Johnson (1990). Presentations of groups. London Mathematical Society Student Texts. 15. Cambridge University Press. pp. 149–150. ISBN 0-521-37203-8.
Dummit and Foote, Abstract Algebra