A domain is a field if it has a common multiple $\!\neq\! 0$ of all elements $\!\neq\! 0$

Suppose $b \ne 0$.

Then it is a unit since $b^2k = b$ for some $k$ so $bk = 1$ by cancellation.

Now let $a \in D$, nonzero. Then $ak = b$ so, multiplying by the inverse of $b$ we get that $a$ is a unit.

Thus $D$ is a field.

Lemma $ $ An abelian monoid is a group $\!\iff\!$ it has a cancellable common multiple $\,b\,$ of all elts.

Proof $\ \ (\Leftarrow)\ \ ab\mid b\underset{{\rm cancel}\ b}\Longrightarrow a\mid 1\,\Rightarrow\, a\,$ invertible. $\ \ (\Rightarrow) \ $ Choose $\,b = 1$.