algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...

, an augmentation ideal is an

ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...

that can be defined in any

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 the set of elements of the give ...

. If ''G'' is a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

and ''R'' a

commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...

, there is a

ring homomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition preservi ...

\varepsilon

, called the

augmentation map In algebra, an augmentation of an associative algebra ''A'' over a commutative ring ''k'' is a ''k''-algebra homomorphism A \to k, typically denoted by ε. An algebra together with an augmentation is called an augmented algebra. The kernel of the ...

, from the group ring

R /math> to R, defined by taking a (finite When constructing , we restrict  to only finite (formal) sums) 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(r) = r for any element r\in R, and \varepsilon is then extended to a homomorphism of ''R''-

module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Mo ...

s in the obvious way. The augmentation ideal is the

kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...

\varepsilon

and is therefore a

two-sided ideal In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers pre ...

in ''R'' 'G'' is generated by the differences

g - 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 In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology loo ...

, amongst other applications.

Examples of quotients by the augmentation ideal

* Let ''G'' a group and

\mathbb /math> the group ring over the integers. Let ''I'' denote the augmentation ideal of \mathbb /math>. 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 \mathbb /math> - module. The coinvariants of ''V'' can then be described as the quotient of ''V'' by ''IV'', where ''I'' is the augmentation ideal in \mathbb /math>. 
* Another class of examples of augmentation ideal can be the

of the

counit In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagram ...

\varepsilon

of any

Hopf algebra Hopf is a German surname. Notable people with the surname include: *Eberhard Hopf (1902–1983), Austrian mathematician *Hans Hopf (1916–1993), German tenor *Heinz Hopf (1894–1971), German mathematician *Heinz Hopf (actor) (1934–2001), Swedis ...

Notes

References

* {{cite book , author=D. L. Johnson , title=Presentations of groups , series=London Mathematical Society Student Texts , volume=15 , publisher=

Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing hou ...

, year=1990 , isbn=0-521-37203-8 , pages=149–150 *Dummit and Foote, Abstract Algebra Ideals (ring theory) Hopf algebras