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