algebra Algebra () is one of the areas of mathematics, 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 mathem ...

, an augmentation of an

associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...

''A'' over a commutative ring ''k'' is a ''k''-

algebra homomorphism In mathematics, an algebra homomorphism is a homomorphism between two associative algebras. More precisely, if and are algebras over a field (or commutative ring) , it is a function F\colon A\to B such that for all in and in , * F(kx) = kF ...

A \to k

, typically denoted by ε. An algebra together with an augmentation is called an augmented algebra. The kernel of the augmentation is a two-sided

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 ...

called the

augmentation ideal 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 /math> to R, defin ...

of ''A''. For example, if

A =k /math> is the group algebra of a finite group''G'', then
: A \to k,\, \sum a_i x_i \mapsto \sum a_i is an augmentation. 

If ''A'' is a

graded algebra In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the ...

which is connected, i.e.

A_0=k

, then the homomorphism

A\to k

which maps an element to its homogeneous component of degree 0 is an augmentation. For example, :

to k, \sum a_ix^i \mapsto a_0

is an augmentation on the

polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variable ...

k /math>.

References

* Algebras {{algebra-stub