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.