Ring homomorphism
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In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication.
More precisely, if R and S are rings, then a ring homomorphism is a function f : R → S such that
- f(a + b) = f(a) + f(b) for all a and b in R
- f(ab) = f(a) f(b) for all a and b in R
- f(1) = 1
(If one does not require rings to have a multiplicative identity then the last condition is dropped.)
The composition of two ring homomorphisms is a ring homomorphism. It follows that the class of all rings forms a category with ring homomorphisms as the morphisms.
Properties
Directly from these definitions, one can deduce:
- f(0) = 0
- f(−a) = −f(a)
- If a has a multiplicative inverse in R, then f(a) has a multiplicative inverse in S and we have f(a−1) = (f(a))−1. Therefore, f induces a group homomorphism from the group of units of R to the group of units of S.
- The kernel of f, defined as ker(f) = {a ∈ R : f(a) = 0} is an ideal in R. Every ideal in R arises from some ring homomorphism in this way. f is injective if and only if the ker(f) = {0}.
- If f is bijective, then its inverse f−1 is also a ring homomorphism. f is called an isomorphism in this case, and the rings R and S are called isomorphic. From the standpoint of ring theory, isomorphic rings cannot be distinguished.
- If Rp is the smallest subring contained in R and Sp is the smallest subring contained in S, then every ring homomorphism f : R → S induces a ring homomorphism fp : Rp → Sp. This can sometimes be used to show that between certain rings R and S, no ring homomorphisms R → S can exist.
- For every ring R, there is a unique ring homomorphism Z → R. This says that the ring of integers is an initial object in the category of rings.
Examples
- The function f : Z → Zn, defined by f(a) = [a]n = a mod n is a surjective ring homomorphism with kernel nZ (see modular arithmetic).
- There is no ring homomorphism Zn → Z for n > 1.
- If R[X] denotes the ring of all polynomials in the variable X with coefficients in the real numbers R, and C denotes the complex numbers, then the function f : R[X] → C defined by f(p) = p(i) (substitute the imaginary unit i for the variable X in the polynomial p) is a surjective ring homomorphism. The kernel of f consists of all polynomials in R[X] which are divisible by X2 + 1.
- If f : R → S is a ring homomorphism between the commutative rings R and S, then f induces a ring homomorphism between the matrix rings Mn(R) → Mn(S).