In ring theory, a branch of abstract algebra, 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 such that f is Additive inverses and the additive identity are part of the structure too, but it is not necessary to require explicitly that they too are respected, because these conditions are consequences of the three conditions above. On the other hand, neglecting to include the condition f = 1S would cause several of the properties below to fail. If in addition f is a bijection, then its inversef−1 is also a ring homomorphism. In this case, f is called a ring isomorphism, and the rings R and S are called isomorphic. From the standpoint of ring theory, isomorphic rings cannot be distinguished. If R and S are rngs, then the natural notion is that of a rng homomorphism, defined as above except without the third condition f = 1S. It is possible to have a rng homomorphism between rings that is not a ring homomorphism. 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. In particular, one obtains the notions of ring endomorphism, ring isomorphism, and ring automorphism.
Properties
Let be a ring homomorphism. Then, directly from these definitions, one can deduce:
If there exists a ring homomorphism then the characteristic of Sdivides the characteristic of R. This can sometimes be used to show that between certain rings R and S, no ring homomorphisms can exist.
If Rp is the smallest subring contained in R and Sp is the smallest subring contained in S, then every ring homomorphism induces a ring homomorphism.
For every ring R, there is a unique ring homomorphism, where 0 denotes the zero ring. This says that the zero ring is a terminal object in the category of rings.
Examples
The function, defined by is a surjective ring homomorphism with kernel nZ.
The function defined by is a rng homomorphism, with kernel 3Z6 and image 2Z6.
If R and S are rings, the zero function from R to S is a ring homomorphism if and only if S is the zero ring. On the other hand, the zero function is always a rng homomorphism.
If R denotes the ring of all polynomials in the variable X with coefficients in the real numbersR, and C denotes the complex numbers, then the function defined by is a surjective ring homomorphism. The kernel of f consists of all polynomials in R which are divisible by.
If is a ring homomorphism between the rings R and S, then f induces a ring homomorphism between the matrix rings.
Given a product of rings, the natural inclusion is not a ring homomorphism ; this is because the map does not send the multiplicative identity of to that of, namely.
The category of rings
Endomorphisms, isomorphisms, and automorphisms
A ring endomorphism is a ring homomorphism from a ring to itself.
A ring isomorphism is a ring homomorphism having a 2-sided inverse that is also a ring homomorphism. One can prove that a ring homomorphism is an isomorphism if and only if it is bijective as a function on the underlying sets. If there exists a ring isomorphism between two rings R and S, then R and S are called isomorphic. Isomorphic rings differ only by a relabeling of elements. Example: Up to isomorphism, there are four rings of order 4. On the other hand, up to isomorphism, there are eleven rngs of order 4.
A ring automorphism is a ring isomorphism from a ring to itself.
Monomorphisms and epimorphisms
Injective ring homomorphisms are identical to monomorphisms in the category of rings: If is a monomorphism that is not injective, then it sends some r1 and r2 to the same element ofS. Consider the two maps g1 and g2 from Z to R that map x to r1 and r2, respectively; and are identical, but since f is a monomorphism this is impossible. However, surjective ring homomorphisms are vastly different from epimorphisms in the category of rings. For example, the inclusion is a ring epimorphism, but not a surjection. However, they are exactly the same as the strong epimorphisms.
