Let $R$ be an integral domain and $r, s$ $\in$ $R$

Question

Let R be an integral domain and r, s $\in$ R.

I know that an integral domain is a commutative ring with identity that has no zero-divisors. Hence, R is a commutative ring with identity and has no zero-divisors.

Given this, can the gcd(r, s) have more than one gcd? Also, would the gcd(r, s) be associates of one another?

I am trying to work this out in $\mathbb{Z}$[$x$] (since I know that $\mathbb{Z}$ is an integral domain as $\mathbb{Z}$[$x$] is a commutative polynomial ring with identity that has no zero-divisors) but I do not know if this is a smart integral domain to work with. Would appreciate some guidance. Thanks in advance

Answer 1

Hint: if $\,d_1,d_2$ are gcds then by definition $\, c\mid d_1\!\iff c\mid a,b\iff c\mid d_2\,$ so $\,\color{#c00}{d_1\mid d_2\mid d_1}$

Beware $ $ In rings that are not domains $\rm\color{#c00}{associates}$ need not be a unit multiples.