  Let $M$ be a cancellative monoid. It is known~\cite{Ta54} that if $M$ is left
amenable then the monoid ring $K[M]$ satisfies Ore condition, that is, there
exist nontrivial common right multiples for the elements of this ring.
In~\cite{Don10} Donnelly shows that a partial converse to this statement is
true. Namely, if the monoid $\mathbb Z^{+}[M]$ of all elements of $\mathbb
Z[M]$ with positive coefficients has nonzero common right multiples, then $M$
is left amenable. He asks whether the converse is true for this particular
statement.
  We show that the converse is false even for the case of groups. If $M$ is a
free metabelian group, then $M$ is amenable but the Ore condition fails for
$\mathbb Z^{+}[M]$. Besides, we study the case of the monoid $M$ of positive
elements of R.\,Thompson's group $F$. The amenability problem for it is a
famous open question. It is equivalent to left amenability of the monoid $M$.
We show that for this case the monoid $\mathbb Z^{+}[M]$ does not satisfy Ore
condition. That is, even if $F$ is amenable, this cannot be shown using the
above sufficient condition.
