# Greatest Common Divisors in Principal Ideal Domain are Associates

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## 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:

\(\ds y_1\) | \(\divides\) | \(\ds y_2\) | as $y_2$ is a greatest common divisor | |||||||||||

\(\ds y_2\) | \(\divides\) | \(\ds y_1\) | as $y_1$ is a greatest common divisor |

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.

$\blacksquare$

## Sources

- 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 62.4$ Factorization in an integral domain: $\text{(ii)}$