On $z^\circ$-ideals in $C(X)$

Azarpanah, F.; Karamzadeh, O. A. S.; Aliabad, A. R.

doi:10.4064/fm_1999_160_1_1_15_25

Cited by 36 publications

(6 citation statements)

References 8 publications

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“…Obviously, if Y = Max(R), then the concepts of H Y -ideal, strong H Y -ideal and Y -Hilbert ideal coincide with the concepts of z-ideal, sz-ideal and Hilbert ideal in the literature, respectively, see [3] and [19]. Also, if Y = Min(R), then the concepts of H Y -ideal and strong H Y -ideal coincide with the concepts of z • -ideal (also known as d-ideal) and sz • -ideal (also known as ξ-ideal), respectively, see [3], [9], [10], [7], [15], [18]. Finally if Y = Spec(R), then the concepts of H Y -ideal, strong H Y -ideal, Y -Hilbert ideals and semiprime ideal coincide.…”

Section: Preliminariesmentioning

confidence: 93%

“…Then this concept studied more generally for the commutative rings, in [18], as an ideal I of R that whenever two elements of R are contained in the same family of maximal ideals and I contains one of them, then it follows that I contains the other one. If we use (Z(f )) • ⊆ (Z(g)) • instead of the above inclusion relation and the minimal prime ideals instead of the maximal ideals in the above definitions, then we obtain the concept of z • -ideal (d-ideal) in C(X) and the commutative rings, which are introduced and carefully studied in [9,10,15]. The concepts of z-ideal and z • -ideal can be generalized to the concepts of sz-ideal and sz • -ideal (ξ-ideal), respectively, based on the finite subsets of the ideals instead of the single points in the ideal, and are studied in [3,7,18].…”

Section: Introductionmentioning

confidence: 99%

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An extension of z-ideals and z^0-ideals

Aliabad¹,

Badie²,

Nazari³

2018

Preprint

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Let R be a commutative ring, Y ⊆ Spec(R) and h Y (S) = {P ∈ Y : S ⊆ P }, for every S ⊆ R. An ideal I is said to be an H Y -ideal whenever it follows from h Y (a) ⊆ h Y (b) and a ∈ I that b ∈ I. A strong H Y -ideal is defined in the same way by replacing an arbitrary finite set F instead of the element a. In this paper these two classes of ideals (which are based on the spectrum of the ring R and are a generalization of the well-known concepts semiprime ideal, zideal, z • -ideal (d-ideal), sz-ideal and sz • -ideal (ξ-ideal)) are studied. We show that the most important results about these concepts, "Zariski topology", "annihilator" and etc can be extended in such a way that the corresponding consequences seems to be trivial and useless. This comprehensive look helps to recognize the resemblances and differences of known concepts better.

show abstract

Section: Preliminariesmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

An extension of z-ideals and z^0-ideals

Aliabad¹,

Badie²,

Nazari³

2018

Preprint

View full text Add to dashboard Cite

show abstract

“…Obviously, if Y = Max(R), then the concepts of H Y -ideal, strong H Y -ideal and Y -Hilbert ideal coincide with the concepts of z-ideal, sz-ideal and Hilbert ideal in the literature, respectively, see [3] and [19]. Also, if Y = Min(R), then the concepts of H Y -ideal and strong H Y -ideal coincide with the concepts of z • -ideal (also known as d-ideal) and sz • -ideal (also known as ξ-ideal), respectively, see [3,7,9,10,15,18]. Finally if Y = Spec(R), then the concepts of H Y -ideal, strong H Y -ideal, Y -Hilbert ideals and semiprime ideal coincide.…”

Section: H Y -Ideals H Y -Filters Strong H Y -Ideals Y -Hilbert Ideals and Their Characterizationsmentioning

confidence: 93%

“…Also, if Y = Min(R), then the concepts of I H and I SH coincide with the concepts of I z • (also known as I • and I • ) and I sz • (also known as ζ(I)-ideal), respectively. We refer to [3,9,10,18], for more information about these concepts.…”

Section: Certain (Strong) H Y -Ideals Over or Contained In An Idealmentioning

confidence: 99%

“…Then this concept was studied more generally for the commutative rings, in [18], as an ideal I of R that whenever two elements of R are contained in the same family of maximal ideals and I contains one of them, then it follows that I contains the other one. If we use (Z(f )) • ⊆ (Z(g)) • instead of the above inclusion relation and the minimal prime ideals instead of the maximal ideals in the above definitions, then we obtain the concept of z • -ideal (d-ideal) in C(X) and the commutative rings, which are introduced and carefully studied in [9,10,15]. The concepts of z-ideal and z • -ideal can be generalized to the concepts of sz-ideal and sz • -ideal (ξ-ideal), respectively, based on the finite subsets of the ideals instead of the single points in the ideal, and are studied in [3,7,18].…”

Section: Introductionmentioning

confidence: 99%

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An extension of $z$-ideals and $z^\circ$-ideals

Badie

Aliabad

Nazari

2020

Hacettepe Journal of Mathematics and Statistics

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Let R be a commutative with unity, Y ⊆ Spec(R), and h Y (S) = {P ∈ Y : S ⊆ P }, for every S ⊆ R. An ideal I is said to be an H Y-ideal whenever it follows from h Y (a) ⊆ h Y (b) and a ∈ I that b ∈ I. A strong H Y-ideal is defined in the same way by replacing an arbitrary finite set F instead of the element a. In this paper these two classes of ideals (which are based on the spectrum of the ring R and are a generalization of the well-known concepts semiprime ideal, z-ideal, z •-ideal (d-ideal), sz-ideal and sz •-ideal (ξideal)) are studied. We show that the most important results about these concepts, "Zariski topology", "annihilator", and etc. can be extended in such a way that the corresponding consequences seems to be trivial and useless. This comprehensive look helps to recognize the resemblances and differences of known concepts better.

show abstract

On z-elements of multiplicative lattices

Goswami,

Dube

2024

Algebra Univers.

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The aim of this paper is to investigate further properties of z-elements in multiplicative lattices. We utilize z-closure operators to extend several properties of z-ideals to z-elements and introduce various distinguished subclasses of z-elements, such as z-prime, z-semiprime, z-primary, z-irreducible, and z-strongly irreducible elements, and study their properties. We provide a characterization of multiplicative lattices where z-elements are closed under finite products and a representation of z-elements in terms of z-irreducible elements in z-Noetherian multiplicative lattices.

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On $z^\circ$-ideals in $C(X)$

Cited by 36 publications

References 8 publications

An extension of z-ideals and z^0-ideals

An extension of z-ideals and z^0-ideals

An extension of $z$-ideals and $z^\circ$-ideals

On z-elements of multiplicative lattices

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