No abstract

No abstract

No abstract

Abstract. It is proved that every abelian VNL-ring is an SVNL-ring, which gives a positive answer to a question of Osba et al. [7]. Some characterizations of duo VNLrings are given and some main results of Osba et al. [7] on commutative VNL-rings are extended to right duo VNL-rings and even abelian GVNL-rings.2000 Mathematics Subject Classification. 16D25, 16E50. Introduction.Rings considered are associative with identity unless the contrary is stated explicitly. An element a of a ring R is π -regular if there exist a positive integer n and x ∈ R such that a n = a n xa n . In the case of n = 1, a is regular. A ring R is π -regular (regular) if every element in R is π -regular (regular). A ring R is an exchange ring if for every a ∈ R there exists an idempotent e ∈ R such that e ∈ aR and (1 − e) ∈ (1 − a)R. A ring R is right (left) duo if every right (left) ideal is two-sided, and R is duo if R is right and left duo. A ring R is abelian if all idempotents are contained in the center and a ring R is reduced if it does not contain nonzero nilpotent elements.Following Contessa [4], a commutative ring R is a VNL-ring if for every a ∈ R, at least one of a or 1 − a is regular. According to Osba et al. [7], a commutative ring R is an SVNL-ring if whenever (S) = R for some nonempty subset S of R, at least one of the elements in S is regular, where (S) is an ideal generated by S. Some properties of VNL-rings and SVNL-rings are investigated in Osba et al. [7]. But they are unable to solve the question whether every VNL-ring is an SVNL-ring. Because of this they are unable to characterize VNL-rings abstractly in the sense of relating them to more familiar classes of rings. In the present paper we define a noncommutative ring R to be a VNL-ring (GVNL-ring) if for every a ∈ R at least one of a or 1 − a is regular (π -regular) and a ring R is an SVNL-ring, if whenever (S) r = R for some nonempty subset S of R, at least one element in S is regular, where (S) r is a right ideal generated by S. The main purpose of this paper is to prove that every abelian VNL-ring is an SVNL-ring, giving an answer to the question of Osba et al. [7] in the affirmative. Furthermore we give some characterizations of duo VNL-rings, and extend and improve some main results of Osba et al. [7] on commutative VNL-rings.Throughout this paper we use the symbol J(R) to denote the Jacobson radical of a ring R, Id(R) its set of idempotents, and Max(R) its maximal spectrum. The left annihilator of an element a in R is denoted by A l (a). For an integer n ≥ 2, the symbol

No abstract