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 : RS 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) = {aR : 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}. Note that in general, for rings with identity the kernel of a ring homomorphism is not a subring since it will not contain the multiplicative identity.
  • The image of f, im(f), is a subring of S.
  • 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 : RS induces a ring homomorphism fp : RpSp. This can sometimes be used to show that between certain rings R and S, no ring homomorphisms RS can exist.
  • If R is a field, then f is either injective or f is the zero function.
  • If both R and S are fields, then im(f) is a subfield of S (if f is not the zero function).
  • If R and S are commutative and S has no zero divisors, then ker(f) is a prime ideal of R (if ker(f) is not trivial).
  • For every ring R, there is a unique ring homomorphism ZR. This says that the ring of integers is an initial object in the category of rings.

Examples

  • The function f : ZZn, 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 ZnZ 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 : RS 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).

Types of ring homomorphisms

  • An injective ring homomorphism is called ring monomorphism.
  • A surjective ring homomorphism is called ring epimorphism.
  • A bijective ring homomorphism is called ring isomorphism.

See also

zh:环的同态与同构