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
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.
If A and B are two unital algebras, then an algebra homomorphism F : A → B is said to be unital if it maps the unity of A to the unity of B. Often the words "algebra homomorphism" are actually used to mean "unital algebra homomorphism", in which case non-unital algebra homomorphisms are excluded.
A unital algebra homomorphism is a (unital) ring homomorphism.