ABSTRACT
Up to now we have examined the relationship between rings by looking at properties they might have in common. Some pairs of rings can actually be placed into a rather closer relationship by means of a function between them. The most important example of this idea is the relationship between Z and Zn, as given by the residue function ϕ : Z→ Zn defined by ϕ(m) = [m]n. Another example is the evaluation function ψ : Q[x] → Q defined by ψ(f) = f(a) (where a is some fixed rational number). Exploring the general context of these examples will then give us a new tool with which to study rings. Because we deal extensively with functions in this chapter, we should
remind you of some terminology regarding them. If R and S are sets and ϕ : R → S is a function, we call R the domain of the function and S its range. Please note that this is a dramatically different use of the word domain than we have already encountered (where domain is really a short version of integral domain), but standard usage forces this ambiguity upon us; we hope you will be able to tell from context which meaning is intended.
