Authors:Ebrahim Hashemi, Fatemeh Shokuhifar, and Abdollah Alhevaz
The intersection of all maximal right ideals of a near-ring N is called the quasi-radical of N. In this paper, first we show that the quasi-radical of the zero-symmetric near-ring of polynomials R0[x] equals to the set of all nilpotent elements of R0[x], when R is a commutative ring with Nil (R)2 = 0. Then we show that the quasi-radical of R0[x] is a subset of the intersection of all maximal left ideals of R0[x]. Also, we give an example to show that for some commutative ring R the quasi-radical of R0[x] coincides with the intersection of all maximal left ideals of R0[x]. Moreover, we prove that the quasi-radical of R0[x] is the greatest quasi-regular (right) ideal of it.
Authors:Abdullah Alahmari, Falih A. Aldosray, and Mohamed Mabrouk
for Mathematics and its Applications , Mathematical Sciences Institute, The Australian National University , 1989 , pp. 61 - 96 .  Dales , H. G. and Zelazko , W. , Generators of maximalleftideals in Banach algebras , Studia Math