# Representing the GCD as linear combination in non-PID rings

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## Abstract

We prove the following fact: If finitely 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 the p i's can also be expressed as a linear combination of p 1,...,p n. We prove an analogous statement in general commutative rings. © 2013 Akadémiai Kiadó, Budapest, Hungary.

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