ring homomorphism - Everything2.com

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).

isomorphism theorems	quotient ring	Chinese remainder theorem	Frobenius endomorphism
integers modulo n	pointless topology	category theory	Automorphism
Ring	irreducibility of cyclotomic polynomials	bijective	field
ring of invisibility	nodal cubic curve	the correspondence between closed sets for the Zariski topology and radical ideals in the polynomial ring	Proof that the algebraic closure of a field is unique
algebraically closed	Proof of the uniqueness of splitting fields	field of fractions	rational functions in one variable
field extension	Polynomial	isomorphism	isomorphic