Strongly fixed ideals in $ C (L)$ and compact frames

Estaji, Ali Akbar; Feizabadi, Abolghasem Karimi; Abedi, Mostafa

doi:10.5817/am2015-1-1

Cited by 4 publications

(3 citation statements)

References 9 publications

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“…Recall from [7] that for α ∈ RL, Z(α) = {p ∈ ΣL : α[p] = 0} is called the zero-set of α. For every A ⊆ RL, we write [7,8]). Note that the intersection of an arbitrary family of strongly z-ideals is a strongly z-ideal.…”

Section: Preliminariesmentioning

confidence: 99%

Sums of Strongly z-Ideals and Prime Ideals in ${mathcal{R}} L$

Estaji

Feizabadi

Sarpoushi

2020

IJMSI

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It is well known that the sum of two z-ideals in C(X) is either C(X) or a z-ideal. The main aim of this paper is to study the sum of strongly z-ideals in RL, the ring of real-valued continuous functions on a frame L. For every ideal I in RL, we introduce the biggest strongly zideal included in I and the smallest strongly z-ideal containing I, denoted by I sz and Isz, respectively. We study some properties of I sz and Isz. Also, it is observed that the sum of any family of minimal prime ideals in the ring RL is either RL or a prime strongly z-ideal in RL. In particular, we show that the sum of two prime ideals in RL which are not chains is a prime strongly z-ideal.

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Section: Preliminariesmentioning

confidence: 99%

Sums of Strongly z-Ideals and Prime Ideals in ${mathcal{R}} L$

Estaji

Feizabadi

Sarpoushi

2020

IJMSI

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“…They are distinct for distinct points. By [14,Lemma 4.2], if p is a prime element of L, then (1) I is a z c -ideal.…”

Section: Remark 33 It Is Evident That For a Familymentioning

confidence: 99%

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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“…[11]). If p is a prime element of a frame L and M p = {α ∈ RL : α[p] = 0} = ker p, then M p is a maximal ideal.…”

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confidence: 99%