Algebra homomorphism

In mathematics, an algebra homomorphism is a homomorphism between two associative algebras. More precisely, if and are algebras over a field (or commutative ring) , it is a function formula_1 such that for all in and in ,
The first two conditions say that is a "K"-linear map (or "K"-module homomorphism if "K" is a commutative ring), and the last condition says that is a (non-unital) ring homomorphism.
If admits an inverse homomorphism, or equivalently if it is bijective, is said to be an isomorphism between and .
Unital algebra homomorphisms.
If "A" and "B" are two unital algebras, then an algebra homomorphism formula_5 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.