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,
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". Every algebra homomorphism is a homomorphism of K-modules.