Combining Local and Von Neumann Regular Rings

Osba, Emad Abu; Henriksen, Melvin; Alkam, Osama

doi:10.1081/agb-120037405

Cited by 17 publications

(28 citation statements)

References 7 publications

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“…We shall see however that an ideal of RL is a z-ideal (as to be defined "topologically") if and only if it is a z-idealà An ideal I of a ring A is a z-idealà la Mason if whenever M(a) ⊇ M(b) and b ∈ I, then a ∈ I. Now here are the characterizations of regular rings that we shall need- (2) and (3) are culled from [20]; (4), (5) and (6) from [27]; and (7) from [1]. Proposition 3.2.…”

Section: P -Framesmentioning

confidence: 99%

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Concerning P-frames, essential P-frames, and strongly zero-dimensional frames

Dube

2009

Algebra Univers.

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We give characterizations of P -frames, essential P -frames and strongly zero-dimensional frames in terms of ring-theoretic properties of the ring of continuous real-valued functions on a frame. We define the m-topology on the ring RL and show that if L belongs to a certain class of frames properly containing the spatial ones, then L is a P -frame iff every ideal of RL is m-closed. We define essential P -frames (analogously to their spatial antecedents) and show that L is a proper essential Pframe iff all the nonmaximal prime ideals of RL are contained in one maximal ideal. Further, we show that L is strongly zero-dimensional iff RL is a clean ring, iff certain types of ideals of RL are generated by idempotents.

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Section: P -Framesmentioning

confidence: 99%

“…A Von Neumann inverse (abbreviated VN-inverse) of a ∈ A is an element b such that a = a 2 b. Elements of A that have VNinverses have the following neat characterizations established in [1]:…”

Section: P -Framesmentioning

confidence: 99%

Concerning P-frames, essential P-frames, and strongly zero-dimensional frames

Dube

2009

Algebra Univers.

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“…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.…”

mentioning

confidence: 94%

“…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).…”

mentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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On Noncommutative VNL-Rings and GVNL-Rings

Chen¹,

Tong²

2006

Glasgow Math. J.

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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

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S-almost semisimple rings

Bouziri

2024

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

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Combining Local and Von Neumann Regular Rings

Cited by 17 publications

References 7 publications

Concerning P-frames, essential P-frames, and strongly zero-dimensional frames

Concerning P-frames, essential P-frames, and strongly zero-dimensional frames

On Noncommutative VNL-Rings and GVNL-Rings

S-almost semisimple rings

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