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.