How do I see that $a = bu$ for some unit $u$?

Question

Suppose elements $a$ and $b$ in a domain satisfy $a \mid b$ and $b \mid a$. How do I see that $a = bu$ for some unit $u$?

Answer 1

Since $a|b$ we can write $ac=b$ for some $c$, and since $b|a$ we can write $bd=a$ for some $d$. Therefore $$ bdc=b $$ or $b(1-dc)=0$. Since the ring is a domain, this implies that either $b=0$ or $dc=1$. If $b=0$ then $a=0$ also, and otherwise it follows that $c$ and $d$ are units.

Answer 2

In a domain you have the cancellation rule:

If $a\ne0$ and $ab=ac$, then $b=c$.