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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

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