Augmentation ideal explained

In algebra, an augmentation ideal is an ideal that can be defined in any group ring.

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

The augmentation ideal is the kernel of

\varepsilon and is therefore a two-sided ideal in R[''G''].

is generated by the differences

g - g' of group elements. Equivalently, it is also generated by \{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

Z [G] the group ring over the integers. Let I denote the augmentation ideal of Z [G] . Then the quotient 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

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 C [G] . \varepsilon of any Hopf algebra.

References

• Book: D. L. Johnson . Presentations of groups . London Mathematical Society Student Texts . 15 . . 1990 . 0-521-37203-8 . 149–150 .
• Dummit and Foote, Abstract Algebra

Notes and References

1. When constructing, we restrict to only finite (formal) sums