In mathematics, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if R and S are rings, then a ring homomorphism is a function f:R \to S that preserves addition, multiplication and multiplicative identity; that is, for all a,b in R. These conditions imply that additive inverses and the additive identity are preserved too. If in addition is a bijection, then its inverse −1 is also a ring homomorphism. In this case, is called a ring isomorphism, and the rings R and S are called isomorphic. From the standpoint of ring theory, isomorphic rings have exactly the same properties. If R and S are rngs, then the corresponding notion is that of a rng homomorphism, defined as above except without the third condition f(1R) = 1S. A rng homomorphism between (unital) rings need not be a ring homomorphism. The composition of two ring homomorphisms is a ring homomorphism. It follows that the rings forms a category with ring homomorphisms as morphisms (see Category of rings). In particular, one obtains the notions of ring endomorphism, ring isomorphism, and ring automorphism.

Properties

Let f : R → S be a ring homomorphism. Then, directly from these definitions, one can deduce: f(a)−1 = f(a−1) . In particular, f induces a group homomorphism from the (multiplicative) group of units of R to the (multiplicative) group of units of S (or of im(f)). Moreover,

Examples

Non-examples

Category of rings

Endomorphisms, isomorphisms, and automorphisms

Monomorphisms and epimorphisms

Injective ring homomorphisms are identical to monomorphisms in the category of rings: If f : R → S is a monomorphism that is not injective, then it sends some r1 and r2 to the same element of S. Consider the two maps g1 and g2 from Z[x] to R that map x to r1 and r2, respectively; f ∘ g1 and f ∘ g2 are identical, but since is a monomorphism this is impossible. However, surjective ring homomorphisms are vastly different from epimorphisms in the category of rings. For example, the inclusion Z ⊆ Q is a ring epimorphism, but not a surjection. However, they are exactly the same as the strong epimorphisms.

Citations