Can a unit divide a unit in a UFD?

Question

My thought process so far is:

Suppose $R$ is a UFD.

Let $a, b$ be units in $R$.

$a | b$ if $b = ac$ for some $c$ in $R$

So if $c$ exists then $a | b$.

$a^{-1}b = c$ so $c$ exists.

QED.

Is this correct?