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 ...

, an augmentation of an associative

''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, define ...

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 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,
: k 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 variables ...

k /math>.

References

* Algebras {{algebra-stub