Amenability of semigroups and the Ore condition for semigroup rings

Abstract

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.