Skew Hurwitz Series Rings and Modules With Beachy-Bliar Conditions

Rajendra Kumar Sharma   ; Amit Bhooshan Singh

Kragujevac Journal of Mathematics, Tome 47 (2023) no. 4, p. 511

Résumé

A ring $R$ satisfies the right Beachy-Blair condition if for every faithful right ideal $J$ of a ring $R$ (that is, a right ideal $J$ of a ring $R$ is faithful if $r_R(J)=0$) is co-faithful (that is, a right ideal $J$ of a ring $R$ is called co-faithful if there exists a finite subset $J_{1}\subseteq J$ such that $r_R(J_{1})=0$). In this note, we prove two main results. \begin{enumerate} em Let $R$ be a ring which is skew Hurwitz series-wise Armendariz, $\omega$-compatible and torsion-free as a $\mathbb{Z}$-module, and $\omega$ be an automorphism of $R$. If $R$ satisfies the right Beachy-Blair condition then the skew Hurwitz series ring $(HR, \omega)$ satisfies the right Beachy-Blair condition. em Let $M_{R}$ be a right $R$-module which is $\omega$-Armendariz of skew Hurwitz series type and torsion-free as a $\mathbb{Z}$-module, and $\omega$ be an automorphism of $R$. If $M_{R}$ satisfies the right Beachy-Blair condition then the skew Hurwitz series module $HM_{(HR, \omega)}$ satisfies the right Beachy-Blair condition. \end{enumerate}

Détail

Classification : 16D25 16S36
Keywords: ring with right Beachy-Blair condition, right R-module with right Beachy-Blair condition, skew Hurwitz series ring, skew Hurwitz series module, reduced ring, $\omega$-compatible ring, $\omega$-compatible module, $\omega$-reduced module