Algebra homomorphism

In mathematics, an algebra homomorphism is a homomorphism between two algebras. More precisely, if A and B are algebras over a field (or a ring) K, it is a function F : A → B such that, for all k in K and x, y in A, one has

• ${\displaystyle F(kx)=kF(x)}$
• ${\displaystyle F(x+y)=F(x)+F(y)}$
• ${\displaystyle F(xy)=F(x)F(y)}$

The first two conditions say that F is a K-linear map, and the last condition says that F preserves the algebra multiplication. So, if the algebras are associative, F is a rng homomorphism, and, if the algebras are rings and F preserves the identity, it is a ring homomorphism.

If F admits an inverse homomorphism, or equivalently if it is bijective, F is said to be an isomorphism between A and B.