Algebra over a field

:This article is about a particular kind of vector space. For other uses of the term "algebra" see algebra (disambiguation).

Definitions

To be precise, let K be a field, and let A be a vector space over K.

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Suppose we are given a binary operation A×A→A, with the result of this operation applied to the vectors x and y in A written as xy.

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Suppose further that the operation is bilinear, i.e.:

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• (x + y)z = xz + yz;
• x(y + z) = xy + xz;
• (ax)y = a(xy); and
• x(by) = b(xy);

for all scalars a and b in K and all vectors x, y, and z.

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Then with this operation, A becomes an algebra over K, and K is the base field of A. The operation is called "multiplication"; note the absence of associativity.

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In general, xy is the product of x and y, and the operation is called multiplication. However, the operation in several special kinds of algebras goes by different names.

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Algebras can also more generally be defined over any commutative ring K: we need a module A over K and a bilinear multiplication operation which satisfies the same identities as above; then A is a K-algebra, and K is the base ring of A.

Related Topics:
Commutative ring - Module

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Two algebras A and B over K are isomorphic if there exists a bijective K-linear map f : A → B such that f(xy) = f(x) f(y) for all x,y in A. For all practical purposes, isomorphic algebras are identical; they just differ in the notation of their elements.

Related Topics:
Bijective - Linear map

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