Algebra homomorphism

An algebra homomorphism between $R$ -algebras $A$ and $B$ (with structure maps $\iota_A\colon R\to A$ , $\iota_B\colon R\to B$ ) is a ring homomorphism $\varphi\colon A\to B$ such that $\varphi\circ \iota_A=\iota_B$ .

Remarks

Algebra homomorphisms are the morphisms in the category of $R$ -algebras; they preserve both multiplication and the base-ring scalars.

Equivalent characterizations

When $R,A,B$ are unital and the structure maps are unital, this is equivalent to requiring $\varphi$ to be a unital ring homomorphism that is $R$ -linear for the induced module structures.

Examples

For any $R$ -algebra $A$ and $a\in A$ , evaluation $R[x]\to A$ , $p(x)\mapsto p(a)$ , is an $R$ -algebra homomorphism from the polynomial ring.
If $J\triangleleft A$ is an ideal stable under the $R$ -algebra structure, then the quotient map $A\to A/J$ is an $R$ -algebra homomorphism.