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 We first show that if R is an -compatible ring and Nil(R) is a locally nilpotent ideal of R, then the set of all nilpotent elements of the near-ring forms an ideal and Then we characterize the unit elements, the regular elements, the π-regular elements and the clean elements of where the base ring R is a semicommutative ring with Also, we prove that the set of all π-regular elements of 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.