Research Article
BibTex RIS Cite
Year 2024, Volume: 36 Issue: 36, 16 - 28, 12.07.2024
https://doi.org/10.24330/ieja.1438744

Abstract

References

  • D. D. Anderson and M. Bataineh, Generalizations of prime ideals, Comm. Algebra, 36(2) (2008), 686-696.
  • D. D. Anderson and M. Winders, Idealization of a module, J. Commut. Algebra, 1(1) (2009), 3-56.
  • M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, 1969.
  • A. Badawi and B. Fahid, On weakly 2-absorbing $\delta$-primary ideals of commutative rings, Georgian Math. J., 27(4) (2020), 503-516.
  • D. E. Dobbs, A. El Khalfi and N. Mahdou Trivial extensions satisfying certain valuation-like properties, Comm. Algebra, 47(5) (2019), 2060-2077.
  • M. Ebrahimpour and R. Nekooei, On generalizations of prime ideals, Comm. Algebra, 40(4) (2012), 1268-1279.
  • S. Glaz, Commutative Coherent Rings, Lecture Notes in Math., 1371, Springer-Verlag, Berlin, 1989.
  • J. A. Huckaba, Commutative Rings with Zero Divisors, Monogr. Textbooks Pure Appl. Math., 117, Marcel Dekker, Inc., New York, 1988.
  • R. Mohamadian, $r$-ideals in commutative rings, Turkish J. Math., 39(5) (2015), 733-749.
  • U. Tekir, S. Koc and K. H. Oral, $n$-ideals of commutative rings, Filomat, 31(10) (2017), 2933-2941.
  • D. Zhao, $\delta$-primary ideals of commutative rings, Kyungpook Math. J., 41(1) (2001), 17-22.

$\delta (0)$-Ideals of Commutative Rings

Year 2024, Volume: 36 Issue: 36, 16 - 28, 12.07.2024
https://doi.org/10.24330/ieja.1438744

Abstract

Let $R$ be a commutative ring with nonzero identity, let $\I (R)$ be the set of all
ideals of $R$ and $\delta : \I (R)\rightarrow\I (R) $ be a function. Then $\delta$ is called an expansion function of ideals of $R$ if whenever $L, I, J$ are ideals of $R$ with $J \subseteq I$, we have $L \subseteq\delta(L)$ and $\delta(J)\subseteq\delta(I)$. In this paper, we present the concept of $\dt$-ideals in commutative rings. A proper ideal $I$ of $R$ is called a $\dt$-ideal if whenever $a$, $b$ $\in R$ with $ab\in I$ and $a\notin \delta (0)$, we have $b\in I$.
Our purpose is to extend the concept of $n$-ideals to $\dt$-ideals of commutative
rings. Then we investigate the basic properties of $\dt$-ideals and also, we
give many examples about $\dt$-ideals.

References

  • D. D. Anderson and M. Bataineh, Generalizations of prime ideals, Comm. Algebra, 36(2) (2008), 686-696.
  • D. D. Anderson and M. Winders, Idealization of a module, J. Commut. Algebra, 1(1) (2009), 3-56.
  • M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, 1969.
  • A. Badawi and B. Fahid, On weakly 2-absorbing $\delta$-primary ideals of commutative rings, Georgian Math. J., 27(4) (2020), 503-516.
  • D. E. Dobbs, A. El Khalfi and N. Mahdou Trivial extensions satisfying certain valuation-like properties, Comm. Algebra, 47(5) (2019), 2060-2077.
  • M. Ebrahimpour and R. Nekooei, On generalizations of prime ideals, Comm. Algebra, 40(4) (2012), 1268-1279.
  • S. Glaz, Commutative Coherent Rings, Lecture Notes in Math., 1371, Springer-Verlag, Berlin, 1989.
  • J. A. Huckaba, Commutative Rings with Zero Divisors, Monogr. Textbooks Pure Appl. Math., 117, Marcel Dekker, Inc., New York, 1988.
  • R. Mohamadian, $r$-ideals in commutative rings, Turkish J. Math., 39(5) (2015), 733-749.
  • U. Tekir, S. Koc and K. H. Oral, $n$-ideals of commutative rings, Filomat, 31(10) (2017), 2933-2941.
  • D. Zhao, $\delta$-primary ideals of commutative rings, Kyungpook Math. J., 41(1) (2001), 17-22.
There are 11 citations in total.

Details

Primary Language English
Subjects Algebra and Number Theory
Journal Section Articles
Authors

Mohamed Chhiti This is me

Bayram Ali Ersoy

Khalid Kaıba

Ünsal Tekir

Early Pub Date February 17, 2024
Publication Date July 12, 2024
Published in Issue Year 2024 Volume: 36 Issue: 36

Cite

APA Chhiti, M., Ersoy, B. A., Kaıba, K., Tekir, Ü. (2024). $\delta (0)$-Ideals of Commutative Rings. International Electronic Journal of Algebra, 36(36), 16-28. https://doi.org/10.24330/ieja.1438744
AMA Chhiti M, Ersoy BA, Kaıba K, Tekir Ü. $\delta (0)$-Ideals of Commutative Rings. IEJA. July 2024;36(36):16-28. doi:10.24330/ieja.1438744
Chicago Chhiti, Mohamed, Bayram Ali Ersoy, Khalid Kaıba, and Ünsal Tekir. “$\delta (0)$-Ideals of Commutative Rings”. International Electronic Journal of Algebra 36, no. 36 (July 2024): 16-28. https://doi.org/10.24330/ieja.1438744.
EndNote Chhiti M, Ersoy BA, Kaıba K, Tekir Ü (July 1, 2024) $\delta (0)$-Ideals of Commutative Rings. International Electronic Journal of Algebra 36 36 16–28.
IEEE M. Chhiti, B. A. Ersoy, K. Kaıba, and Ü. Tekir, “$\delta (0)$-Ideals of Commutative Rings”, IEJA, vol. 36, no. 36, pp. 16–28, 2024, doi: 10.24330/ieja.1438744.
ISNAD Chhiti, Mohamed et al. “$\delta (0)$-Ideals of Commutative Rings”. International Electronic Journal of Algebra 36/36 (July 2024), 16-28. https://doi.org/10.24330/ieja.1438744.
JAMA Chhiti M, Ersoy BA, Kaıba K, Tekir Ü. $\delta (0)$-Ideals of Commutative Rings. IEJA. 2024;36:16–28.
MLA Chhiti, Mohamed et al. “$\delta (0)$-Ideals of Commutative Rings”. International Electronic Journal of Algebra, vol. 36, no. 36, 2024, pp. 16-28, doi:10.24330/ieja.1438744.
Vancouver Chhiti M, Ersoy BA, Kaıba K, Tekir Ü. $\delta (0)$-Ideals of Commutative Rings. IEJA. 2024;36(36):16-28.