On clean and regular elements of noncommutative ring extensions

Abstract

Let R be an associative ring with identity and α be an endomorphism of R. In this article, we are interested to study the some of relations between a ring R and that of D. A. Jordan’s construction of the ring A(R,α) as well as the skew Laurent polynomial ring $R[x,x^{-1};\alpha ]$. The main propose of this article is to characterize the unit elements, the idempotent elements, von Neumann regular elements, π-regular elements, von Neumann local elements and also the clean elements of the skew Laurent polynomial ring as well as Jordan’s construction of the ring A(R,α). Applying these characterizations, one can easily get some nice radical-theoretic properties of the mentioned classes of rings.

Keywords:

• Idempotent element
• regular element
• unit element
• skew Laurent polynomial rings
• Jordan construction

2010 Mathematics Subject Classification:

• Primary: 16S36
• Secondary: 16U60
• 16U99

Acknowledgements

The authors would like to thank the referee for a careful reading of the paper and for all of the constructive comments, which have greatly improved the presentation of the article.