Principal Rings and their Invariant Factors

Abstract

In this article we put forward a new look at the theory of principal and Invariant-Factor rings, with a view toward facilitating the formalization, automation, and archiving of results and their proofs. We take an elementary and constructive approach: standard techniques such as prime ideals and factorization of elements are avoided, and determinant constructions are minimized. Using such ‘‘computationally friendly’’ methods, the main existence and uniqueness results on invariant factors for a f.g. torsion module are derived, and several new algebraic constructions and results are found. The lattice of principal integral ideals for any commutative Bezoutian ring is explicitly constructed based on a first-order proof overlooked in the literature, together with a proof that this lattice is distributive. A ‘‘Lagrange quotient’’ theorem for finitely generated modules over any principal ring is stated for the first time. A very constructive new proof is given that a principal ring has the Hermite property, so is also an Invariant-Factor ring. A calculus that is needed in the ideal lattice, naturally yields a number of formulas valid for a function lattice.