Results on Reversible Properties in Skew Monoid Rings

Abstract

This paper introduces the concept of \(\sigma\)-skew strongly \(M\)-nil-reversible rings as a unified generalization of strongly \(M\)-reversible and \(M\)-compatible rings. We then examine the reversibility properties of certain monoid ring extensions. Specifically, for an \(M\)-Armendariz and \(\sigma\)-skew Armendariz ring \(R\), we show that \(R\) is \(\sigma\)-skew strongly \(M\)-nil-reversible. Furthermore, we establish necessary conditions under which the skew monoid ring \(R * M\) inherits this property, particularly when \(R\) is an \(NI\)-ring and \(M\) is a \(u.p.\)-monoid. Additionally, we investigate the persistence of this condition under factor rings (over strictly ordered monoids) and classical right quotient rings (over Ore rings). Finally, an explicit counterexample of \(2 \times 2\) matrices over a field \(F\) is provided to show that this property does not hold in general.