Greatest Common Divisors in Principal Ideal Domain are Associates

From ProofWiki
Jump to navigation Jump to search

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