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

Dube, Themba

doi:10.1007/s00012-009-0006-2

Cited by 38 publications

(12 citation statements)

References 19 publications

(39 reference statements)

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“…We recall the notation z-ideal of a ring A as was introduced by Mason in [25]. We refer to z-ideals as defined in [25] as "z-idealsá la In [8,Corollary 3.8], Dube shows that an ideal of RL is a z-ideal if and only if it is a z-idealá la Mason. Here we introduce and study z c -ideals in R c L. We begin by below definition.…”

Section: Z C -Ideals In R C Lmentioning

confidence: 99%

“…This algebraic definition of z-ideal was coined in the context of rings of continuous functions by Kohls in [22] and is also in the text Rings of continuous functions by Gillman and Jerison [16]. In pointfree topology, z-ideals were introduced by Dube in [8] in terms of the cozero map. z-Ideals have been studied in the theory of abelian lattice-ordered groups [4,27] and in the context of Riesz space in [17] and [18].…”

Section: Introductionmentioning

confidence: 99%

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zc-ideals and prime ideals in the ring RcL

Estaji¹,

Feizabadi²,

Sarpoushi³

2018

Filomat

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The ring R c L is introduced as a sub-f-ring of RL as a pointfree analogue to the subring C c (X) of C(X) consisting of elements with the countable image. We introduce z c-ideals in R c L and study their properties. We prove that for any frame L, there exists a space X such that βL OX with C c (X) R c (OX) R c βL R * c L, and from this, we conclude that if α, β ∈ R c L, |α| ≤ |β| q for some q > 1, then α is a multiple of β in R c L. Also, we show that IJ = I ∩ J whenever I and J are z c-ideals. In particular, we prove that an ideal of R c L is a z c-ideal if and only if it is a z-ideals. In addition, we study the relation between z c-ideals and prime ideals in R c L. Finally, we prove that R c L is a Gelfand ring.

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Section: Z C -Ideals In R C Lmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

zc-ideals and prime ideals in the ring RcL

Estaji¹,

Feizabadi²,

Sarpoushi³

2018

Filomat

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“…A P -frame is a frame in which every cozero element is complemented. We refer to [16] and [15] for some characterizations of F -frames and P -frames, respectively.…”

Section: Variants Of Roundnessmentioning

confidence: 99%

“…Let I ∈ Pt(βL). By [15,Proposition 3.9], it suffices to show that M I = O I . Consider the quotient map ξ : βL → 2 given by ξ(J) = 0 ⇐⇒ J ≤ I.…”

Section: Corollary 34 An F -Frame Is a P -Frame Iff Every Quotient mentioning

confidence: 99%

Covering maximal ideals with minimal primes

Dube

Ighedo

2015

Algebra Univers.

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A ring is a UMP-ring if every maximal ideal in the ring is the union of the minimal prime ideals it contains. Banerjee, Ghosh and Henriksen have characterized Tychonoff spaces X for which C(X) is a UMP-ring. One of the characterizations is that every singleton of βX is what is called a nearly round subset. In this article, we define nearly round quotient maps, and use them to characterize completely regular frames L for which RL is a UMP-ring. All such frames are almost P -frames, and an Oz-frame is of this kind precisely when it is an almost P -frame. If L is perfectly normal (and hence if L is metrizable), then RL is a UMP-ring if and only if L is Boolean. If A is a UMP-ring which is a Q-algebra, then every ideal of A, when viewed as a ring in its own right, is a UMP-ring.

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“…Our example will be based on special types of frames. The reader will recall that a P-frame is a frame each of whose cozero elements is complemented, and that a frame L is extremally disconnected (or strongly projectable) if x Ã x ÃÃ 1 for each x P L. See [8] for several characterizations of P-frames. However, we shall need the following characterization (which does not appear in [8]) in terms of W-maps.…”

Section: Lemma 43mentioning

confidence: 99%

Contracting the Socle in Rings of Continuous Functions

Dube¹

2010

Rend. Sem. Mat. Univ. Padova

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-Not every ring homomorphism contracts the socle of its codomain to an ideal contained in the socle of its domain. Rarer still does a homomorphism contract the socle to the socle. We find conditions on a frame homomorphism that ensure that the induced ring homomorphism contracts the socle of the codomain to an ideal contained in the socle of the domain. The surjective among these frame homomorphisms induce ring homomorphisms that contract the socle to the socle. These homomorphisms characterize P-frames as those L for which every frame homomorphism with L as domain is of this kind.

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

Cited by 38 publications

References 19 publications

zc-ideals and prime ideals in the ring RcL

zc-ideals and prime ideals in the ring RcL

Covering maximal ideals with minimal primes

Contracting the Socle in Rings of Continuous Functions

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