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.

众所周知，如果使一个可分解的单半体M服从，则该半体半环K [ M ]满足Ore条件，也就是说，该环的元素存在非平凡的公倍数。Donnelly（Semigroup论坛81：389–392，2010）表明，与此说法有部分相反的说法是正确的。也就是说，如果系数为正的\（{\ mathbb {Z}} [M] \）的所有元素的等分体\（{\ mathbb {Z}} ^ {+} [M] \）具有非零的公共右倍数，那么M就顺服了。他询问相反的说法是否适用于此特定声明。我们证明，即使对于群体而言，反之亦然。如果M是一个自由的metabelian组，则M是可以满足的，但是\（{\ mathbb {Z}} ^ {+} [M] \）的矿石条件失败。此外，我们研究了汤普森群F的正元素的等式M的情况。F的顺应性问题是一个著名的开放问题。它等效于类半体M的左适应性。我们证明，在这种情况下，单面体\（{\ mathbb {Z}} ^ {+} [M] \）不满足Ore条件。即，即使F是合适的，也不能使用上述充分条件来表示。