Adequacy of nonsingular matrices over commutative principal ideal domains

TL;DR

This paper extends the concept of adequacy from commutative domains to noncommutative Bézout rings and demonstrates that nonsingular second-order matrices over a commutative principal ideal domain are adequate.

Contribution

It introduces the notion of adequacy in noncommutative Bézout rings and proves adequacy for nonsingular second-order matrices over commutative principal ideal domains.

Findings

Adequacy concept extended to noncommutative Bézout rings

Nonsingular second-order matrices over PID are adequate

Provides theoretical foundation for matrix adequacy in new algebraic contexts

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 B\'ezout rings. We show that the set of nonsingular second-order matrices over a commutative principal ideal domain is adequate.