- Examples of Quotients by the Augmentation Ideal
- Notes
- References
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 
, called the augmentation map, from the group ring 
to R, defined by taking a (finite[1]) sum 
to 
(Here 
and 
.) In less formal terms, 
for any element 
, 
for any element 
, and is then extended to a homomorphism of R-modules in the obvious way.
The augmentation ideal {{mvar|A}} is the kernel of and is therefore a two-sided ideal in R[G].
of group elements. Equivalently, it is also generated by 
, 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
the group ring over the integers. Let I denote the augmentation ideal of . Then the quotient {{math|I/I{{sup|2}} }} 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
- module. The coinvariants of V can then be described as the quotient of V by IV, where I is the augmentation ideal in . - Another class of examples of augmentation ideal can be the kernel of the counit of any Hopf algebra.
Notes
References
- {{cite book | author=D. L. Johnson | title=Presentations of groups | series=London Mathematical Society Student Texts | volume=15 | publisher=Cambridge University Press | year=1990 | isbn=0-521-37203-8 | pages=149–150 }}
- Dummit and Foote, Abstract Algebra
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