s. A function f
is called a ring homomorphism
if it satisfies:
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,
See also isomorphism theorems, quotient ring.