Adequacy of nonsingular matrices over commutative principal ideal domains

TL;DR

This work extends Helmer’s adequacy concept to noncommutative Bézout rings and analyzes when matrix divisors are adequate. It develops a Smith normal form/transformations framework and introduces spectrum-based criteria that characterize left divisors and their common divisors, proving that the set of nonsingular $2\times2$ matrices over a commutative PID is adequate. The paper provides explicit constructions and lemmas (via ${\rm SNF}((A,B)_{l})$ and $\Sigma(\cdot)$) to describe left divisors, and includes concrete examples that illustrate both the strengths of the approach and its distinctions from Gatalevych’s definition. It concludes with a critical comparison showing advantages of the present definition, including stability up to equivalence and constructive descriptions, which have potential algorithmic implications in noncommutative matrix theory.

Abstract

The notion of the adequacy of commutative domains was introduced by Helmer in Bull. Amer.Math. Soc., 49 (1943), 225--236. In the present paper we extend the concept of adequacy to noncommutative Bézout rings. We show that the set of nonsingular second-order matrices over a commutative principal ideal domain is adequate.

Theorems & Definitions (10)

• Theorem 1
• Theorem 2
• Lemma 1
• proof
• proof : Proof of Theorem \ref{['T:2']}. Necessity.
• Lemma 2
• proof
• Lemma 3
• proof
• proof : Proof of Theorem \ref{['T:1']}