Let
R and
S be
rings. A
function f:
R>S is called a
ring homomorphism (of rings
with
1)
if it satisfies:

f(1_{R})=1_{R}

f(a+b)=f(a)+f(b), for all a,b in R

f(ab)=f(a)f(b), for all a,b in R.
If f is also bijective it is called a ring isomorphism and R
and S are said to be isomorphic rings.
For example, consider the polynomial ring k[x] for some
field k (you can take k=R the real numbers
or k=C). Then, after choosing, a in k we get a homomorphism
f:k[x]>k defined by evaluating at a. That is,
f(p(x))=p(a).
See also isomorphism theorems, quotient ring.