Representing the GCD as linear combination in non-PID 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.