Algebra over a field

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

K-algebra morphism

Given K-algebras A and B, a K-algebra morphism is a map f:A o B such that f is a ring morphism that commutes with the scalar multiplication defined by η; that is, the following diagram commutes:

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:egin{matrix}

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&& K && \

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& eta_A swarrow & , & eta_B searrow & \

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A && egin{matrix} f \ longrightarrow end{matrix} && B

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end{matrix}

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which one may write as

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:f(ka)=kf(a)

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for all kin K and a in A. The space of all K-algebra morphisms is frequently written as

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:mathbf{Alg}_K (A,B)

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A K-algebra isomorphism is a bijective K-algebra morphism.

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