Principal Ideal Domains

Last week, Ari taught you about one kind of “simple” (in the nontechnical sense) ring, specifically semisimple rings. These have the property that every module splits as a direct sum of simple modules (in the technical sense). This week, we’ll look at a rather different kind of ring, namely a principal ideal domain, or PID. These rings, like semisimple rings, have the property that every (finitely generated) module is a direct sum of “simple” modules, though here we use simple in the nontechnical sense of “easy to understand.” However, while this property of modules was almost the definition of semisimple rings, for PIDs it is much less obvious, and the bulk of our time will be devoted to proving this classification of modules. This classification is very powerful, and its applications include both a complete classification of finitely generated abelian groups and a classification of matrices up to conjugation over C (or any algebraically closed field). The main example of a PID we will focus on is the integers Z, for which modules are just abelian groups. However, another important example will be k[x], the ring of polynomials in one variable over a field. Indeed, while k[x] and Z may look like fairly different rings at first, they are in fact very similar just by both being PIDs. The first, most obvious difference between what I will do and what Ari did is that PIDs are by definition commutative. Thus throughout these notes, all rings will be assumed to be commutative. If a, b, c . . . ∈ R are elements of a ring, we let (a, b, c, . . .) denote the ideal they generate. More generally, if a, b, c . . . ∈ M are elements of an R-module, we let (a, b, c, . . .) denote the submodule they generate.