# Does $(a)=(b)$ imply that $a$ and $b$ are associate in a principal ideal ring?

Let $R$ be a commutative principal ideal ring with identity. Suppose that $a,b\in R$ and $(a)=(b)$. I'd like to know if there always exists $u\in R^\times$ such that $a=bu$.

I know several counterexamples if $R$ is not a principal ideal ring, so I'm a bit curious about whether it holds for principal ideal rings.

• Usually people talk about $PID$, principal ideal domains. These are integral domains, so from $a=xb$ and $b=ya$ you can establish that $a(1-xy)=0$ so that indeed $x,y$ are units. Is there any reason to study non-domains? – Blah Mar 30 '15 at 7:49