Ring homomorphism

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 f : R → S such that f is: addition preserving: for all a and b in R, multiplication preserving: for all a and b in R, and unit (multiplicative identity) preserving: 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. If in addition f is a bijection, then its inverse f−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 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 class of all rings forms a with ring homomorphisms as the morphisms (cf. the ). In particular, one obtains the notions of ring endomorphism, ring isomorphism, and ring automorphism. Let be a ring homomorphism. Then, directly from these definitions, one can deduce: f(0R) = 0S. f(−a) = −f(a) for all a in R. For any unit element a in R, f(a) is a unit element such that 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)). The of f, denoted im(f), is a subring of S. The kernel of f, defined as ker(f) = , is an ideal in R. Every ideal in a ring R arises from some ring homomorphism in this way. The homomorphism f is injective if and only if ker(f) = . If there exists a ring homomorphism f : R → S then the characteristic of S divides the characteristic of R.