On clean and regular elements of near-ring of skew polynomials

Abstract

In this paper, we are interested to investigate some properties of the zero-symmetric near-ring of skew polynomials $R_0[x;\alpha ,\delta ].$ We first show that if R is an $(\alpha ,\delta )$-compatible ring and Nil(R) is a locally nilpotent ideal of R, then the set of all nilpotent elements of the near-ring $R_0[x;\alpha ,\delta ]$ forms an ideal and $\mathit{Nil}(R_0[x;\alpha ,\delta ])=\mathit{Nil}{(R)}_0[x].$ Then we characterize the unit elements, the regular elements, the π-regular elements and the clean elements of $R_0[x;\alpha ,\delta ],$ where the base ring R is a semicommutative ring with $\mathit{Nil}{(R)}^2=0.$ Also, we prove that the set of all π-regular elements of $R_0[x;\alpha ,\delta ]$ forms a semigroup. These results are somewhat surprising since, in contrast to the skew polynomial ring case, the near-ring of skew polynomials has substitution for its “multiplication” operation.

Keywords:

• Clean elements
• Near-ring
• Nilpotent elements
• π-regular elements
• Regular elements
• Skew polynomial ring
• Unit elements

AMS Subject Classification:

• 16Y30
• 16U60

Acknowledgements

We are very grateful to the referee for his/her useful comments, which helped us to improve the paper.