On skew-reversible endomorphisms

Abstract

For a ring R with a skew-reversible endomorphism $\sigma$, we prove the following: (i) $W(R)=L(R)=N^{*}(R)=\mathit{\text{Nil}}(R)={\mathit{\text{Nil}}}_{\sigma }(R)=N_{\sigma }(R)=L_{\sigma }(R)=W_{\sigma }(R)$, (ii) $R/{\mathit{\text{Nil}}}_{\sigma }(R)$ is a $\overline{\sigma }$-rigid ring where $\overline{\sigma }(r+{\mathit{\text{Nil}}}_{\sigma }(R))=\sigma (r)+{\mathit{\text{Nil}}}_{\sigma }(R)$ for $r\in R$, (iii) If $\sigma$ is an automorphism then we have $W(R[x;\sigma ])$ $=W(R)[x;\sigma ]$ $=L(R[x;\sigma ])$ $=N^{*}(R[x;\sigma ])$, and $W(R[x,x^{-1};\sigma ])=W(R)[x,x^{-1};\sigma ]=L(R[x,x^{-1};\sigma ])$ $=N^{*}(R[x,x^{-1};\sigma ])$, where $W(-)$, $W_{\sigma }(-)$, $L(-)$, $L_{\sigma }(-)$, $N^{*}(-)$, $N_{\sigma }(-)$, $\mathit{\text{Nil}}(-)$ and ${\mathit{\text{Nil}}}_{\sigma }(-)$ mean the Wedderburn radical, the $\sigma$-Wedderburn radical, the Levitzki radical, the $\sigma$-Levitzki radical, the upper nilradical, the upper $\sigma$-nil radical, the set of all nilpotent elements, and the set of all $\sigma$-nilpotent elements, respectively. In addition, the structure of rings with skew-reversible endomorphisms is studied in relation to near related rings.

KEYWORDS:

• $\sigma$-nil radical
• ($\sigma$-)nilpotent, ($\sigma$-)Wedderburn radical
• skew Laurent polynomial ring
• skew polynomial ring
• skew-reversible endomorphism

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 16N40
• 16W20
• 16S36

Acknowledgments

The authors thank the referee for very careful reading of the manuscript and many valuable suggestions that improved the paper by much.

Additional information

Funding

The first named author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2019R1F1A1040405) and the second named author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2022R1A5A1033624).