1. ## Associates

R is an Integral Domain.
Let c be a greatest common divisor of a and b in R and let f be an associate of c. Show that f is also a greatest common divisor of a and b in R.

Thoughts...
I think i should show that c|f and f|c. Then f would be the gcd as well.
f = uc, where u is a unit, so c|f. Can find another unit v $\in$ R s.t. uv. = 1 (i.e. v is the inverse of u).
Then c = vf, hence f|c. So f is gcd as well?

On second thoughts, since R is an Int Domain maybe im not allowed to assume that uv = 1...

R is an Integral Domain.
Let c be a greatest common divisor of a and b in R and let f be an associate of c. Show that f is also a greatest common divisor of a and b in R.

Thoughts...
I think i should show that c|f and f|c. Then f would be the gcd as well.
f = uc, where u is a unit, so c|f. Can find another unit v $\in$ R s.t. uv. = 1 (i.e. v is the inverse of u).
Then c = vf, hence f|c. So f is gcd as well?

On second thoughts, since R is an Int Domain maybe im not allowed to assume that uv = 1...
If $f=uc,$ suppose $uu'=u'u=1$ for some $u'.$ Now $a=a'c$ and $b=b'c$ for some $a',b'.$ Hence $a=a'u'f$ and $b=b'u'f,$ showing that $f$ divides both $a$ and $b$. Now if $g$ divides both $a$ and $b$ then $g$ divides $c$ as $c$ is a GCD. So $c=g'g$ for some g'. It follows that $f=ug'g$ and so $g$ divides $f.$ This shows that $f$ is a GCD.