Recall the following relevant definitions. We say that
$b$ is divisible by $a$ in $R$, or $a\mid b$ in $R$, if $b = r a$ for some $r\in R$.
$a$ and $b$ are associates in $R$ if $a\mid b$ and $b\mid a$, (or, equivalently, if $aR = bR$).
$u\in R$ is a unit if it has a multiplicative inverse (a $v\in R$ such that $uv=vu=1$).
$a$ and $b$ are unit multiples in $R$ if $a = ub$ for some unit $u\in R$.
Given these definitions, my question is,
If $R$ is a commutative ring with unity and $a,b\in R$ are associates in $R$, are $a$ and $b$ unit multiples in $R$?
I was told that this not always true. But I encountered some difficulties in finding a counterexample.