Algebra Homomorphism

Algebra Homomorphism

A homomorphism between two algebras, A and B, over a field (or ring) K, is a map such that for all k in K and x,y in A,

• F(kx) = kF(x)

• F(x + y) = F(x) + F(y)

• F(xy) = F(x)F(y)

If F is bijective then F is said to be an isomorphism between A and B.

A common abbreviation for "homomorphism between algebras" is "algebra homomorphism" or "algebra map". It is easy to see that every algebra homomorphism is a homomorphism of K-modules.