A GENERALIZATION OF PP-RINGS AND p.q.-BAER RINGS

Liu, Zhongkui; Zhao, Ruyan

doi:10.1017/s0017089506003016

Cited by 30 publications

(5 citation statements)

References 9 publications

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“…In this paper, we continue the studies of [1]- [38]. As an outcome, substantial progresses in the understanding the structure of p.p.…”

mentioning

confidence: 81%

Structure theory of p.p. rings and their generalizations

Tarizadeh¹

2020

Preprint

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In this paper, new and significant advances on the understanding the structure of p.p. rings and their generalizations have been made. Specially among them, it is proved that a commutative ring R is a generalized p.p. ring if and only if R is a generalized p.f. ring and its minimal spectrum is Zariski compact. We also prove that the total ring of fractions of a given ring R is absolutely flat if and only if R is reduced, its minimal spectrum is Zariski compact and every finitely generated and faithful ideal of R admits a non zero-divisor. Some of the major results of the literature either are improved or are proven by new methods. In particular, we give a new and elementary proof to the fact that a commutative ring R is a p.p. ring if and only if R[x] is a p.p. ring. We also prove the same assertion for p.f. rings. Finally, the new notion of fluffy ring is introduced and studied which generalizes at once both generalized p.p. ring and almost p.p. ring notions.

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“…In this paper, we continue the studies of [1]- [38]. As an outcome, substantial progresses in the understanding the structure of p.p.…”

mentioning

confidence: 81%

Structure theory of p.p. rings and their generalizations

Tarizadeh¹

2020

Preprint

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“…Because b r a j ∈ C U R and ec i ∈ eR, so (ϕψC e φ)(s) = b r ω gr (a j ω ρs (ec i )) = 0 since ω is an auto- Proof. If R is a quasi-Baer ring, then it is a left p.q.-Baer ring, as shown in [24]. Therefore, the result can be deduced from Theorem 3.3.…”

mentioning

confidence: 84%

“…Proof. If R is a quasi-Baer ring, it is a left p.q.-Baer ring, as stated in [24]. Consequently, R is strongly CM-reflexive by [10].…”

mentioning

confidence: 98%

On Crossed Product Rings Over p.q.-Baer and Quasi-Baer Rings

Ali

2023

Int. J. Anal. Appl.

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In this paper, we consider a ring R and a monoid M equipped with a twisting map f: M×M -> U(R) and an action map ω: M -> Aut(R). The main objective of our study is to investigate the conditions under which the crossed product structure R⋊M is p.q.-Baer and quasi-Baer rings, and how this property relates to the p.q.-Baer property of R and the existence of a generalized join in I(R) for M-indexed subsets, where I(R) denotes the set of ideals of R. Additionally, we prove a connection between R being a left p.q.-Baer ring and the CM-quasi-Armendariz property. Moreover, we prove that for any element φ2=φ, there exist an idempotent element e2=e such that φ = ce. We then prove that R is quasi-Baer if and only if the crossed product structure R⋊M is quasi-Baer. Finally, we present novel results regarding various constructions for crossed products.

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“…These rings have been studied by various mathematicians in the past years, see e.g. [3], [23], [27], [29], [44, §3] and [45]. We have the following result.…”

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confidence: 87%

Characterizations of Gelfand rings specially clean rings and their dual rings

Aghajani¹,

Tarizadeh²

2018

Preprint

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In this paper, new criteria for zero dimensional rings, Gelfand rings, clean rings and mp-rings are given. The equivalency of some of the classical criteria are also proved by new and simple methods. A new and interesting class of rings is introduced and studied, we call them purified rings. Specially, non-trivial characterizations for reduced purified rings are given. Purified rings are actually the dual of clean rings. The pure ideals of reduced Gelfand rings and mp-rings are characterized. It is also proved that if the topology of a scheme is Hausdorff then the affine opens of that scheme is stable under taking finite unions (and nonempty finite intersections). In particular, every compact scheme is an affine scheme.

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A GENERALIZATION OF PP-RINGS AND p.q.-BAER RINGS

Cited by 30 publications

References 9 publications

Structure theory of p.p. rings and their generalizations

Structure theory of p.p. rings and their generalizations

On Crossed Product Rings Over p.q.-Baer and Quasi-Baer Rings

Characterizations of Gelfand rings specially clean rings and their dual rings

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