Chapter II - RESIDUE RINGS AND RINGS OF QUOTIENTS

Summary

Homomorphisms and isomorphisms. There are associated with a given ring R certain other rings, which can be derived from it by means of algebraic processes. The residue rings of R and the rings of quotients of R are among these derived rings, and they are of particular importance, because with their aid it is possible to simplify some of the problems which have to do with the ideals of R itself. In order to study the relation of R to one of its residue rings, we shall first define what we mean by a homomorphic mapping of R.

Suppose that we have a mapping, σ, of R into R′, where R′ may, at first, be a set of objects of any kind. Thus with each element a ∈ R there is associated a definite element σ(a) of R′. If it happens that every element of R′ is the image of at least one element of R, then we say that σ maps R on to R′. Assume now that an addition and a multiplication have been defined on R′. By this we mean that if a′ and b′ belong to R′, then a′ + b′ and a′b′ have been defined and are again elements of R′; but it is not to be assumed that those relations hold, † which must be satisfied if R′ is to be a ring.