The Reflexive Condition on Skew Monoid Rings

Abstract

This paper is devoted to introducing and studying two concepts, σ-skew strongly M-reflexive and σ-skew strongly M-nil-reflexive on monoid rings, which are generalizations of strongly M-reflexive and M-compatible. The paper covers the basic properties of skew monoid rings of the form R∗M. It is shown that if R is a left AP P (quasi Armendariz, semiprime rings, respectively), then R is σ-skew strongly M-reflexive. Moreover, if R is a NI-ring and M is a u.p-monoid, then R is σ-skew strongly M-nil-reflexive. Additionally, under some necessary and sufficient conditions, a skew monoid ring R ∗ M is proven to be σ-skew strongly M-nil-reflexive when σ : M → Aut(R) is a monoid homomorphism. Furthermore, if R is a left AP P, then the upper triangular matrix ring Tn(R) is σ ̄-skew strongly M-nil-reflexive, where n is a positive integer. Finally, the paper provides some examples and discusses related results from the subject.