Definition:Unital Associative Commutative Algebra Homomorphism

Definition
Let $A$ be a commutative ring with unity.

Let $B$ and $C$ be $A$-algebras.

Definition 1
Let $B$ and $C$ be viewed as rings under $A$, say $(B, f)$ and $(C, g)$.

An $A$-algebra homomorphism $h : B \to C$ is a morphism of rings under $A$.

That is, a unital ring homomorphism $h$ such that $g = h \circ f$:
 * $\xymatrix{

A \ar[d]_f \ar[r]^{g} & C\\ B \ar[ru]_{h} }$

Definition 2
Let $B$ and $C$ be viewed as unital algebras over $A$.

An $A$-algebra homomorphism $h : B \to C$ is a unital algebra homomorphism.

Also see

 * Equivalence of Definitions of Unital Associative Commutative Algebra