Prove that, in a domain, $(a)=(b)$ iff $a = bu$ for some unit $u$.
By $(a)=(b)$, it also means that $a\mid b$ and $b\mid a$ so we can write them as $a=bu$ and $b=av$ for some $u, v \in R$ where $u$ and $v$ are units. In a domain you can cancel nonzero elements and units are nonzero elements so will I use substitution to get something like $a=(av)u$ and domains are cancellative so the $a$ terms will cancel and I will get $0=uv$ which suggests that either u or v must be $0$ but units can't be zero...
I just don't know where to take it from here or if I'm on the right track.