Generating solutions of a linear equation and structure of elements of the Zelisko group

TL;DR

This paper investigates the structure of solutions to linear equations in certain algebraic domains and explores the properties of the Zelisko group, introducing the concept of generating solutions that serve as fundamental elements.

Contribution

It develops a theory of solutions for linear equations in homomorphic images of commutative Bezout domains and analyzes the structure of the Zelisko group using generating solutions.

Findings

Existence of a generating solution that divides all others in the solution set

Generating solutions are pairwise associates

Structural insights into the Zelisko group based on these solutions

Abstract

Solutions of a linear equation b=ax in a homomorphic image of a commutative Bezout domain of stable range 1.5 is developed. It is proved that the set of solutions of a solvable linear equation contains at least one solution that divides the rest, which is called a generating solution. Generating solutions are pairwise associates. Using this result, the structure of elements of the Zelisko group is investigated.