Amenability of semigroups and the Ore condition for semigroup rings

It is known that if a cancellative monoid M is left amenable then the monoid ring K[M] satisfies the Ore condition, that is, there exist nontrivial common right multiples for the elements of this ring. Donnelly (Semigroup Forum 81:389–392, 2010) 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 F 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 the Ore condition. That is, even if F is amenable, this cannot be shown using the above sufficient condition.