Representing the GCD as linear combination in non-PID rings (Q2448971)

In a principal ideal domain the greatest common divisor of two elements \(a\) and \(b\) is a linear combination of \(a\) and \(b\) by definition. As a consequence, the gcd of elements \(p_1,\dots,p_n\) is a linear combination of \(p_1,\dots,p_n\). The author generalizes this fact as follows. Let \(R\) be a unique factorization domain and suppose that \(\gcd(p_i,p_j)\) is a linear combination of \(p_i\) and \(p_j\) for all \(i\) and \(j\); then \(\gcd(p_1,\dots,p_n)\) is a linear combination of \(p_1,\dots,p_n\). The author also proves a result in the same spirit that holds for general commutative rings.