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. If R and S are rings and f : R -> S is a function, we require

  • 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

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 in R : f(a) = 0} is an ideal in R. Every ideal in R arises from some ring homomorphism in this way.
  • 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.
  • The composition of two ring homomorphisms is a ring homomorphism; the class of all rings together with the ring homomorphisms forms a category.

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.
  • 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).