Representing the GCD as linear combination in non-PID rings

TL;DR

This paper proves that in certain rings, the GCD of multiple elements can be expressed as a linear combination if pairwise GCDs can, extending known results from PIDs to non-PID rings and commutative rings.

Contribution

It generalizes the linear combination property of GCDs from pairs to multiple elements in non-PID rings and commutative rings.

Findings

GCD of all elements can be expressed as a linear combination under given conditions

Extension of GCD linear combination property from PIDs to non-PID rings

Analogous results established in commutative rings

Abstract

In this note we prove the following fact: if finite many elements $p_1,p_2,...,p_n$ of a unique factorization domain are given such that the greatest common divisor of each pair $(p_i,p_j)$ can be expressed as a linear combination of $p_i$ and $p_j$ then the greatest common divisor of all $p_i$s also can be expressed as a linear combination of $p_1,...,p_n$. We prove am analogous statement in commutative rings.