A ring homomorphism 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 the operations on the left-hand side is in 
and on the right-hand side in 
. Note that a homomorphism must preserve the additive inverse
map because 
so 
.
A ring homomorphism for unit rings (i.e., rings with a multiplicative identity) satisfies the additional property that one multiplicative
identity is mapped to the other, i.e., 
.
