Ring homomorphism

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{math, number, function}

In ring theory or 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[1]

  • 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

Naturally, 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 (cf. the category of rings).

Contents

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 (multiplicative) group of units of R to the (multiplicative) 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 a commutative ring R arises from some ring homomorphism in this way. For rings with identity, the kernel of a ring homomorphism is a subring without identity.
  • The homomorphism f is injective if and only if the ker(f) = {0}.
  • 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 there exists a ring homomorphism f : RS then the characteristic of S divides the characteristic of R. This can sometimes be used to show that between certain rings R and S, no ring homomorphisms RS can exist.
  • 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.
  • If R is a field, then f is either injective or f is the zero function. Note that f can only be the zero function if S is a trivial ring or if we don't require that f preserves the multiplicative identity.
  • If both R and S are fields (and f is not the zero function), then im(f) is a subfield of S, so this constitutes a field extension.
  • If R and S are commutative and S has no zero divisors, then ker(f) is a prime ideal of R.
  • If R and S are commutative, S is a field, and f is surjective, then ker(f) is a maximal ideal of R.
  • 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.

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