Homomorphism of commutative unital rings

Definition
Let $$R,S$$ be commutative unital rings. A function $$f:R \to S$$ is termed a homomorphism of commutative unital rings, or simply a homomorphism, if it satisfies the following conditions:


 * 1) $$f(a + b) = f(a) + f(b)$$
 * 2) $$f(0) = 0$$
 * 3) $$f(-a) - -f(a)$$
 * 4) $$f(1) = 1$$
 * 5) $$f(ab) = f(a)f(b)$$

It turns out that conditions (2) and (3) follow from (1). However, condition (4) does not follow from condition (5). One comes across situations where a map of commutative unital rings preserves the additive and multiplicative structure but does not send the multiplicative identity to the multiplicative identity; such a map is not a homomorphism of commutative unital rings.