Definition 2

Given two algebras E, E' over a commutative ring A, a homomorphism of E into E' is a mapping $f : E \to E'$ such that (1) f is an A*-module homomorphism*; (2) $f(xy) = f(x)f(y)$ for all $x \in E$ and $y \in E$.

Clearly the composition of two A-algebra homomorphisms is an A-algebra homomorphism. Every bijective algebra homomorphism is an isomorphism. Therefore A-algebra homomorphisms may be taken as morphisms of the species of A-algebra structure (Set Theory, IV, 2, no. 1). We shall always suppose in what follows that this choice of morphisms has been made. If E, E' are two A-algebras, let $\operatorname{Hom}_{A\text{-alg}}(E, E')$ denote the set of A-algebra homomorphisms of E into E'.

Let E, E' be two algebras each with a unit element. A homomorphism of E into E' mapping the unit element of E to the unit element of E' is called a unital homomorphism (or unital algebra morphism).