Algebra homomorphism over a field
= Algebra homomorphism over a field
= Algebra homomorphism
{synonym}
= k-algebra homomorphism
{synonym}
For a <field> $k$, a homomorphism of $k$-algebras is a unital <ring homomorphism> preserving scalar multiplication by every element of $k$. For unital algebras it is equivalently a <ring> homomorphism commuting with their structural maps from $k$. Pullback by a <morphism of algebraic varieties> gives such a homomorphism between their <coordinate rings>.