Homomorphism of commutative unital rings

From Commalg

Definition

Let be commutative unital rings. A function is termed a homomorphism of commutative unital rings, or simply a homomorphism, if it satisfies the following conditions:

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.