On Skew Strongly Reversible Rings Relative to a Monoid

Abstract:

      For a monoid $M$, we introduce the concept of skew strongly $M$-reversible rings which is a generalization of strongly $M$-reversible rings, and investigate their properties. It is shown that if $G$ is a finitely generated Abelian group, then $G$ is torsion-free if and only if there exists a ring $R$ with $|R| \geq 2$ such that $R$ is skew strongly $G$-reversible. Moreover, we prove that if $R$ is a right Ore ring with classical right quotient ring $Q$, then $R$ is skew strongly $M$-reversible if and only if $Q$ is skew strongly $M$-reversible.