Principal Ideal Domains and Their Modules

Abstract

An integral domain in which every ideal is generated by a single element is called a principle ideal domain or PID. Finitely generated modules over a PID are completely classified in this chapter. They are uniquely determined by a collection of ring elements called the elementary divisors. This theory is applied to two of the most prominent PIDs in mathematics: the ring of integers, \({\mathbb Z}\), and the polynomial rings F[x], where F is a field. In the case of the integers, the theory yields a complete classification of finitely generated abelian groups. In the case of the polynomial ring one obtains a complete analysis of a linear transformation of a finite-dimensional vector space. The rational canonical form, and, by enlarging the field, the Jordan form, emerge from these invariants.