Principal rings and their invariant factors (Q1422206)

In this article, the author examines one topic from commutative algebra related to rings whose finitely presented modules may be given as a sum of cyclic modules. The purpose is to examine some of the fundamental results of the theory from the point of view of economy of logical development and algorithmic implementation. In section 2, the author reviews diagonalization algorithms for a f.\,p. module over a principal ideal domain. In section 4, the author establishes that any principal ideal ring is a Hermite ring, and thus the results of this section will also apply to any principal ideal ring. The author presents two diagonalization algorithms which will serve different purposes. By the results of section 5 we know that over a principal ring, two isomorphic f.\,g. torsion modules possess the same capacity ideals. In section 6 we see how this implies that the entire Smith forms are also equal. In appendix A, the author lists some join and meet formulas that are needed, and in appendix B, he shows how these equations correspond to true formulas relating sup and inf for functions, whose domain can be almost anything, into a totally ordered Abelian group.