SavedAdd your journal

Journals / Gazi University Journal of Science Part A: Engineering and Innovation / 2021 / Volume: 8 Issue: 4

On Prime Ideals Related to an Ideal in a Commutative Ring

Ortaç Öneş(AKDENIZ UNIVERSITY)0000-0001-6777-9192
Pages
—
Publication Date
2021-12-30
🏛️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)

Moreover a direct connection between prime ideals related to an ideal and radical ideals is obtained.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:This article focuses on the notion of prime ideal related to an ideal of a commutative ring.
📊
Key Findings & Data:Some of its characteristics are studied and the relations between ideals and prime ideals related to these ideals in a commutative ring are examined.
💡
Conclusion & Impact:Moreover a direct connection between prime ideals related to an ideal and radical ideals is obtained.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Prime Ideal · Radical Ideal · Localization

Abstract

This article focuses on the notion of prime ideal related to an ideal of a commutative ring. Some of its characteristics are studied and the relations between ideals and prime ideals related to these ideals in a commutative ring are examined. Then some useful connections among them are obtained. Both similarities and differences between prime ideals related to an ideal and prime ideals are pointed out. Moreover a direct connection between prime ideals related to an ideal and radical ideals is obtained.

🔖

Cite This Article

Öneş, O. (2021). On Prime Ideals Related to an Ideal in a Commutative Ring. Gazi University Journal of Science Part A: Engineering and Innovation, Volume: 8 Issue: 4. https://doi.org/10.54287/gujsa.1013194

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (19)

  1. Anderson, D. D., & Bataineh, M. (2008). Generalizations of prime ideals. Communications in Algebra, 36, 686-696.
  2. Anderson, D. D., & Smith, E. (2003). Weakly prime ideals. Houston Journal of Mathematics, 29, 831-840.
  3. Atiyah, M. F., & MacDonald, I. G. (1969). Introduction to Commutative Algebra. Addison-Wesley.
  4. Ebrahimpour, M., & Nekooei, R. (2012). On generalizations of prime ideals. Communications in Algebra, 40, 1268-1279.
  5. Jenkins, J., & Smith, P.F. (1992). On the prime radical of a module over a commutative ring. Communications in Algebra, 20, 3593-3602.
  6. Kaplansky, I. (1974). Commutative Rings. The University of Chicago.
  7. Kirby, D. (1966). Components of ideals in a commutative ring. Ann. Mat. Pura Appl., 4, 109-125.
  8. Kirby, D. (1969). Closure operations on ideals and submodules. J. London. Math. Soc., 44, 283-291.
  9. Lam, T. Y. (2001). A First Course in Noncommutative rings. Springer.
  10. Lam, T. Y., & Reyes, M. L. (2008). A prime ideal principle in commutative algebra. Journal of Algebra, 319, 3006-3027.
  11. Matsumura, H. (1986). Commutative Ring Theory. Cambridge University Press.
  12. McConnell, J. C., & Robson, J. C. (1987). Noncommutative Noetherian Rings. Wiley Chichester.
  13. Naghipour, A. R. (2005). A simple proof of Cohen’s theorem. The American Mathematical Monthly, 112, 825-826.
  14. Öneş, O. (2019). A generalization of prime ideals in a commutative ring. In: Proceedings Book of the 2nd Mediterranean International Conference of Pure Applied Mathematics and Related Areas, Paris, France, 28-31 August, 124-126.
  15. Öneş, O. (2020). On radical formula in modules over noncommutative rings. Filomat, 34(2), 443-449.
  16. Öneş, O., & Alkan, M. (2017). On the left O-prime ideals over a noncommutative ring. Advanced Studies in Contemporary Mathematics, 27(1), 107-113.
  17. Öneş, O., & Alkan, M. (2018). A note on graded ring with prime spectrum. Advanced Studies in Contemporary Mathematics, 28(4), 625-634.
  18. Öneş, O., & Alkan, M. (2019a). Multiplication modules with prime spectrum. Turkish Journal of Mathematics, 43(4), 2000-2009.
  19. Öneş, O., & Alkan, M. (2019b). Zariski subspace topologies on ideals. Hacettepe Journal of Mathematics and Statistics, 48(6), 1667-1674.