Representing the GCD as linear combination in non-PID rings

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.