Editing Algebra over a field (section)

{{Short description|Vector space equipped with a bilinear product}}
{{Algebraic structures |Algebra}}

In [[mathematics]], an '''algebra over a field''' (often simply called an '''algebra''') is a [[vector space]] equipped with a [[bilinear map|bilinear]] [[product (mathematics)|product]]. Thus, an algebra is an [[algebraic structure]] consisting of a [[set (mathematics)|set]] together with operations of multiplication and addition and [[scalar multiplication]] by elements of a [[field (mathematics)|field]] and satisfying the axioms implied by "vector space" and "bilinear".<ref>See also {{harvnb|Hazewinkel|Gubareni|Kirichenko|2004|p=[{{Google books|AibpdVNkFDYC|plainurl=y|page=3|text=an algebra over a field k}} 3] Proposition 1.1.1}}</ref>

The multiplication operation in an algebra may or may not be [[associative]], leading to the notions of [[associative algebra]]s where associativity of multiplication is assumed, and [[non-associative algebra]]s, where associativity is not assumed (but not excluded, either). Given an integer ''n'', the [[ring (mathematics)|ring]] of [[real matrix|real]] [[square matrix|square matrices]] of order ''n'' is an example of an associative algebra over the field of [[real number]]s under [[matrix addition]] and [[matrix multiplication]] since matrix multiplication is associative. Three-dimensional [[Euclidean space]] with multiplication given by the [[vector cross product]] is an example of a nonassociative algebra over the field of real numbers since the vector cross product is nonassociative, satisfying the [[Jacobi identity]] instead.

An algebra is '''unital''' or '''unitary''' if it has an [[identity element]] with respect to the multiplication. The ring of real square matrices of order ''n'' forms a unital algebra since the [[identity matrix]] of order ''n'' is the identity element with respect to matrix multiplication. It is an example of a unital associative algebra, a [[unital ring|(unital) ring]] that is also a vector space.

Many authors use the term ''algebra'' to mean ''associative algebra'', or ''unital associative algebra'', or in  some subjects such as [[algebraic geometry]], ''unital associative commutative algebra''.

Replacing the field of scalars by a [[commutative ring]] leads to the more general notion of an [[#Generalization: algebra over a ring|algebra over a ring]]. Algebras are not to be confused with vector spaces equipped with a [[bilinear form]], like [[inner product space]]s, as, for such a space, the result of a product is not in the space, but rather in the field of coefficients.