# Definition:Unital Associative Commutative Algebra Homomorphism

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## 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.