SavedAdd your journal

Journals / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 49 Issue: 1

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

Ali Rezaei Aliabad(Shahid Chamran University of Ahvaz)0000-0003-1293-3652
Mehdi Badie(Jundi-Shapur University of Technology)0000-0003-1114-3130
Sajad Nazari(Shahid Chamran University of Ahvaz)0000-0002-4295-2435
Pages
—
Publication Date
2020-02-06
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

This comprehensive look helps to recognize the resemblances and differences of known concepts better.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:Let $R$ be a commutative ring, $Y\subseteq Spec(R)$ and $ h_Y(S)=\{P\in Y:S\subseteq P \}$, for every $S\subseteq R$.
📊
Key Findings & Data:An ideal $I$ is said to be an $\mathcal{H}_Y$-ideal whenever it follows from $h_Y(a)\subseteq h_Y(b)$ and $a\in I$ that $b\in I$.
💡
Conclusion & Impact:This comprehensive look helps to recognize the resemblances and differences of known concepts better.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:$z$-ideal · $z^\circ$-ideal · strong $z$-ideal

Abstract

Let $R$ be a commutative ring, $Y\subseteq Spec(R)$ and $ h_Y(S)=\{P\in Y:S\subseteq P \}$, for every $S\subseteq R$. An ideal $I$ is said to be an $\mathcal{H}_Y$-ideal whenever it follows from $h_Y(a)\subseteq h_Y(b)$ and $a\in I$ that $b\in I$. A strong $\mathcal{H}_Y$-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, $z^{\circ}$-ideal (d-ideal), sz-ideal and $sz^{\circ}$-ideal ($\xi$-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.

🔖

Cite This Article

Aliabad, A. R., Badie, M., Nazari, S. (2020). An extension of $z$-ideals and $z^\circ$-ideals. Hacettepe Journal of Mathematics and Statistics, Volume: 49 Issue: 1. https://doi.org/10.15672/hujms.455030

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (22)

  1. A.R. Aliabad and M. Badie, Fixed-place ideals in commutative rings, Comment. Math. Univ. Carolin. 54 (1), 53–68, 2013.
  2. A.R. Aliabad and M. Badie, On Bourbaki associated prime divisors of an ideal, Quaest. Math. 42 (4), 479-500, 2019.
  3. A.R. Aliabad and R. Mohamadian, On $sz^{\circ}$-ideals in polynomial rings, Comm. Algebra 39 (2) (2011), 701–717, 2011.
  4. A.R. Aliabad, R. Mohamadian, and S. Nazari, On regular ideals in reduced rings, Filomat 31 (12), 3715–3726, 2017.
  5. A.R. Aliabad and S. Nazari, On the spectrum of a commutative ring via C(X), (to appear).
  6. A.R. Aliabad, A. Taherifar, and N. Tayarzadeh, $\alpha$-Baer rings and some related concepts via C(X), Quaest. Math. 39 (3), 401–419, 2016.
  7. G. Artico, U. Marconi, and R. Moresco, A subspace of Spec(A) and its connexions with the maximal ring of quotients, Rend. Sem. Mat. Univ., Padova 64, 93–107, 1981.
  8. M.F. Atiyah and I.G. Macdonald, Introduction to Commutative Algebra, vol. 2, Addison-Wesley Reading, 1969.
  9. F. Azarpanah, O.A.S. Karamzadeh, and A.R. Aliabad, On $z^{\circ}$-ideals in C(X), Fund. Math. 160 (1), 15-25,1999.
  10. F. Azarpanah, O.A.S. Karamzadeh, and A.R. Aliabad, On ideals consisting entirely of zero-divisors, Comm. Algebra 28 (2), 1061–1073, 2000.
  11. T.S. Blyth, Lattices and Ordered Algebraic Structures, Springer-Verlag, London, 2005.
  12. L. Gillman and M. Jerison, Rings of Continuous Functions, Van Nostrand Reinhold, New York, 1960.
  13. G. Grätzer, General Lattice Theory, Birkhäuser, Basel, 1998.
  14. M. Henriksen and M. Jerison, The space of minimal prime ideals of a commutative ring, Trans. Amer. Math. Soc. 115, 110–130, 1965.
  15. C.B. Huijsmans and B. De Patger, On z-ideals and d-ideals in Riesz spaces. I, Indag. Math. (Proceeding) A 83 (2), 183–195, 1980.
  16. D.G. Johnson and M. Mandelker, Functions with pseudocompact support, Topology Appl. 3 (4), 331–338, 1973.
  17. J. Kist, Minimal prime ideals in commutative semigroups, Proc. Lond. Math. Soc. 3 (1), 31–50, 1963.
  18. G. Mason, z-ideals and prime ideals, J. Algebra 26 (2), 280–297, 1973.
  19. G. Mason, Prime ideals and quotient rings of reduced rings, Math. Jpn. 34 (6), 941– 956, 1989.
  20. Y. Quentel, Sur la compacité du spectre minimal d’un anneau, Bull. Soc. Math. France 99, 265–272, 1971.
  21. R.Y. Sharp, Steps in Commutative Algebra, Cambridge university press, London, 1990.
  22. S. Willard, General Topology, Addison Wesley, Reading Mass., New York, 1970.