Augmentation ideal
Encyclopedia
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 sum
to
Here ri is an element of R and gi an element of G. The sums are finite, by definition of the group ring. In less formal terms,
is defined as 1R whatever the element g in G, and
is then extended to a homomorphism of R-module
s in the obvious way. The augmentation ideal is the kernel of
, and is therefore a two-sided ideal in R[G]. It is generated by the differences
of group elements.
Furthermore it is also generated by
which is a basis for the augmentation ideal 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.
Another class of examples of augmentation ideal can be the kernel of the counit
of any Hopf algebra
.
The augmentation ideal plays a basic role in group cohomology
, amongst other applications.
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, an augmentation ideal is an ideal
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
that can be defined in any group ring
Group ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring and its basis is one-to-one with the given group. As a ring, its addition law is that of the free...
. If G is a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
and R a commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
, there is a ring homomorphism
Ring homomorphism
In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication....
, called the augmentation map, from the group ring
to R, defined by taking a sum
to
Here ri is an element of R and gi an element of G. The sums are finite, by definition of the group ring. In less formal terms,
is defined as 1R whatever the element g in G, and
is then extended to a homomorphism of R-moduleModule (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
s in the obvious way. The augmentation ideal is the kernel of
, and is therefore a two-sided ideal in R[G]. It is generated by the differences
of group elements.
Furthermore it is also generated by
which is a basis for the augmentation ideal 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.
Another class of examples of augmentation ideal can be the kernel of the counit
of any Hopf algebraHopf algebra
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an algebra and a coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property.Hopf algebras occur naturally...
.
The augmentation ideal plays a basic role in group cohomology
Group cohomology
In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology is a way to study groups using a sequence of functors H n. The study of fixed points of groups acting on modules and quotient modules...
, amongst other applications.