Definition:Unital Algebra Homomorphism

Definition
Let $R$ be a commutative ring.

Let $\left({A, *}\right)$ and $\left({B, \times}\right)$ be unital algebras over $R$ with units $1_A$ and $1_B$.

A unital algebra homomorphism $f : A \to B$ is a algebra homomorphism such that $f \left({1_A}\right) = 1_B$.