SavedAdd your journal

Journals / Hacettepe Journal of Mathematics and Statistics / 2021 / Cilt: 50 - Sayı: 3

$k$-Congruences and the Zariski topology in semirings

Song-CHOL HAN
Pages
699–709
Publication Date
2022-10-18
DOI
—
🏛️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)

We first prove that the $k$-prime spectrum, i.e. the set of all $k$-prime ideals equipped with the Zariski topology is a spectral space, and then prove that the set of all $k$-prime congruences is homeomorphic to the $k$-prime spectrum with respect to their Zariski topologies.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:The purpose of this paper is to study topological properties of both the set of all $k$-prime ideals and the set of all $k$-prime congruences for any commutative semiring with zero and identity.
📊
Key Findings & Data:The purpose of this paper is to study topological properties of both the set of all $k$-prime ideals and the set of all $k$-prime congruences for any commutative semiring with zero and identity.
💡
Conclusion & Impact:We first prove that the $k$-prime spectrum, i.e. the set of all $k$-prime ideals equipped with the Zariski topology is a spectral space, and then prove that the set of all $k$-prime congruences is homeomorphic to the $k$-prime spectrum with respect to their Zariski topologies.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:semiring · ideal · congruence

Abstract

The purpose of this paper is to study topological properties of both the set of all $k$-prime ideals and the set of all $k$-prime congruences for any commutative semiring with zero and identity. We first prove that the $k$-prime spectrum, i.e. the set of all $k$-prime ideals equipped with the Zariski topology is a spectral space, and then prove that the set of all $k$-prime congruences is homeomorphic to the $k$-prime spectrum with respect to their Zariski topologies.

🔖

Cite This Article

HAN, S. (2022). $k$-Congruences and the Zariski topology in semirings. Hacettepe Journal of Mathematics and Statistics, Cilt: 50 - Sayı: 3, 699–709. https://ajindex.com/articles/529495

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (20)

  1. S.E. Atani and R.E. Atani, A Zariski topology for k-semirings, Quasigroups Related Systems 20, 29-36, 2012
  2. M.F. Atiyah and I.G. Macdonald, Introduction to Commutative Algebra, Addison Wesley, Massachusetts, 1969
  3. F. Callialp, G. Ulucak and U. Tekir, On the Zariski topology over an L-module M, Turkish J. Math. 41, 326-336, 2017
  4. R. El Bashir and T. Kepka, Congruence-simple semirings, Semigroup Forum 75, 588-608, 2007
  5. J.S. Golan, Semirings and their Applications, Kluwer Academic Publishers, Dordrecht, 1999
  6. S.C. Han, Maximal k-ideals and r-ideals in semirings, J. Algebra Appl. 14 (10), 1250195, 13 pages, 2015
  7. U. Hebisch and H.J. Weinert, Semirings: Algebraic Theory and Applications in Computer Science, World Scientific, Singapore, 1998
  8. M. Henriksen, Ideals in semirings with commutative addition, Amer. Math. Soc. Notices 5, 321, 1958
  9. M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142, 43-60, 1969
  10. P. Lescot, Absolute algebra II – Ideals and spectra, J. Pure Appl. Algebra 215, 1782- 1790, 2011
  11. P. Lescot, Absolute algebra III – The saturated spectrum, J. Pure Appl. Algebra 216, 1004-1015, 2012
  12. P. Lescot, Prime and primary ideals in semirings, Osaka J. Math. 52, 721-736, 2015
  13. G. Maze, C. Monico and J. Rosenthal, Public key cryptography based on semigroup actions, Adv. Math. Commun. 1, 489-507, 2007
  14. A. Pena, L.M. Ruza and J. Vielma, Separation axioms and the prime spectrum of commutative semirings, Rev. Notas Mat. 5 (2), 66-82, 2009
  15. M.K. Sen and M.R. Adhikari, On maximal k-ideals of semirings, Proc. Amer. Math. Soc. 118, 699-703, 1993
  16. H.E. Stone, Matrix representation of simple halfrings, Trans. Amer. Math. Soc. 233, 339-353, 1977
  17. H.S. Vandiver, Note on a simple type of algebra in which the cancellation law of addition does not hold, Bull. Amer. Math. Soc. 40, 916-920, 1934
  18. H.J. Weinert, M.K. Sen and M.R. Adhikari, One-sided k-ideals and h-ideals in semirings, Math. Pannon. 7 (1), 147-162, 1996
  19. G. Yesilot, On prime and maximal k-subsemimodules of semimodules, Hacet. J. Math. Stat. 39, 305-312, 2010
  20. B. Zhou and W. Yao, Relations between ideals and regular congruences in idempotent semirings with a zero, Basic Sci. J. Textile Univ. 24, 253-255, 2011.