ABSTRACT
Up to now we have mostly examined the relationship between rings by looking at properties they might have in common. But 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). A third more general example is the idea of ring isomorphism we introduced in Chapter 7: A bijection between two rings that preserves addition and multiplication. Exploring the general context of these examples will then give us a new tool with which to study rings.
