Greatest Common Divisors in Principal Ideal Domain are Associates

Theorem
Let $\struct {D, +, \circ}$ be a principal ideal domain.

Let $S = \set {a_1, a_2, \dotsc, a_n}$ be a set of non-zero elements of $D$.

Let $y_1$ and $y_2$ be greatest common divisors of $S$.

Then $y_1$ and $y_2$ are associates.

Proof
From Finite Set of Elements in Principal Ideal Domain has GCD we have that at least one such greatest common divisor exists.

So, let $y_1$ and $y_2$ be greatest common divisors of $S$.

Then:

Thus we have:
 * $y_1 \divides y_2$ and $y_2 \divides y_1$

where $\divides$ denotes divisibility.

Hence the result, by definition of associates.