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 $\varphi\circ \iota_A=\iota_B$. Equivalently, $\varphi$ is a unital ring homomorphism that is $R$-linear as a module homomorphism with respect to the induced $R$-module structures coming from the algebra structure

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

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.