MODULES OVER A PID A module over a PID is an abelian group

A module over a PID is an abelian group that also carries multiplication by a particularly convenient ring of scalars. Indeed, when the scalar ring is the integers, the module is precisely an abelian group. This writeup presents the structure theorem for finitely generated modules over a PID. Although the result is essentially similar to the theorem for finitely generated abelian groups from earlier in the course, the result for modules over a PID is not merely generality for its own sake. Its first and best known application bears on linear algebra, and in this context the PID is the polynomial ring k[X] (k a field) rather than the integer ring Z. In this case the theorem instantly gives the rational canonical form and the Jordan canonical form of a linear transformation, results that are tedious to establish by more elementary methods. The next writeup of the course will give this application. This writeup’s proof of the structure theorem for finitely generated modules over a PID is not the argument found in many texts. That argument, which confused me for a long time, is a detail-busy algorithm that • proceeds from an assumption that often is left tacit, that the module has a presentation, meaning a characterizing description in terms of generators and relations that fully determines it (we so assumed in our proof of the theorem for finitely generated abelian groups); • in fact is not really an algorithm after all unless the PID carries algorithmic arithmetic (by contrast, the argument here foregrounds its use of the axiom of choice); • blurs the distinction between two different uniqueness questions in a way that is all too easily lost on the student.