On Bourbaki associated prime divisors of an ideal

Aliabad, A. R.; Badie, Mehdi

doi:10.2989/16073606.2018.1459924

Cited by 3 publications

(3 citation statements)

References 7 publications

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“…Example 5.9. Suppose that R = C(N), Y = B(R) and M is a maximal ideal of R. Since the zero ideal of R is a fixed-place ideal, by [2,Theorem 4.7], it follows that there is an…”

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

confidence: 99%

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

“…Example 5.9. Suppose that R = C(N), Y = B(R) and M is a maximal ideal of R. Since the zero ideal of R is a fixed-place ideal, by [2,Theorem 4.7], it follows that there is an…”

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

confidence: 99%

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

Badie

Aliabad

Nazari

2020

Hacettepe Journal of Mathematics and Statistics

View full text Add to dashboard Cite

show abstract

“…Example 5.9. Suppose that R = C(N), Y = B(R) and M is a maximal ideal of R. Since the zero ideal of R is a fixed-place ideal, by [1,Theorem 4.7], it follows that there is an ultrafilter…”

Section: Some Important Classes Of H Y -Ideals Strong H Y -Ideals And...mentioning

confidence: 99%

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

“…We use B(R) instead of B({0}). For more information about the fixed-place ideals and anti fixed-place ideals, see [6,7].…”

Section: Introductionmentioning

confidence: 99%

Notes on the zero-divisor graph and annihilating-ideal graph of a reduced ring

Badie

2021

Analele Universitatii "Ovidius" Constanta - Seria Matematica

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We translate some graph properties of 𝔸𝔾(R) and Γ(R) to some topological properties of Zariski topology. We prove that the facts “(1) The zero ideal of R is an anti fixed-place ideal. (2) Min(R) does not have any isolated point. (3) Rad(𝔸𝔾 (R)) = 3. (4) Rad(Γ(R)) = 3. (5) Γ(R) is triangulated (6) 𝔸𝔾 (R) is triangulated.” are equivalent. Also, we show that if the zero ideal of a ring R is a fixed-place ideal, then dt t (𝔸𝔾 (R)) = |ℬ(R)| and also if in addition |Min(R)| > 2, then dt(𝔸𝔾 (R)) = |ℬ (R)|. Finally, it is shown that dt(𝔸𝔾 (R)) is finite if and only if dt t (𝔸𝔾 (R)) is finite if and only if Min(R) is finite.

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On Bourbaki associated prime divisors of an ideal

Cited by 3 publications

References 7 publications

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

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

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

Notes on the zero-divisor graph and annihilating-ideal graph of a reduced ring

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