A ring homomorphism, also called a ring morphism, is a map 
between two rings such that
1. Addition is preserved: 
,
2. The zero element is mapped to zero: 
, and
3. Multiplication is preserved: 
,
where 
and the operations on the left-hand side are in 
and on the right-hand side in 
. Note that a homomorphism
must preserve the additive inverse map
because 
so 
.
Under the identity-preserving convention, a ring homomorphism between unit rings (i.e., rings with a multiplicative
identity) satisfies the additional property that one multiplicative
identity is mapped to the other, i.e., 
.
