Let $R$ and $S$ each be a D24: Ring.
A D18: Map $f : R \to S$ is a ring homomorphism from $R$ to $S$ if and only if

(1)	$\forall \, x, y \in R : f(x + y) = f(x) + f(y)$ (D948: Group homomorphism)
(2)	$\forall \, x, y \in R : f(x y) = f(x) f(y)$ (D2189: Semigroup homomorphism)