<b><i>N</i></b>-prime Ideal Graph of a Finite Commutative Ring

Kalamani Duraisamy; Hariharan Murugan

doi:doi:10.11648/j.acm.20261506.11

Research Article |

| Peer-Reviewed

N-prime Ideal Graph of a Finite Commutative Ring

Kalamani Duraisamy

, Hariharan Murugan^*

Published in Applied and Computational Mathematics (Volume 15, Issue 6)

Received: 30 September 2026 Accepted: 12 October 2026 Published: 5 November 2026

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

In this paper, we introduce and investigate the N-prime ideal graph of a finite commutative ring with unity. The concept of prime ideal graphs over rings has been studied by many authors. Let R be a finite commutative ring with unity and let J be a proper ideal of R. The N-prime ideal graph of R with respect to J, denoted by G_J, is the simple undirected graph whose vertex set consists of the elements of the ring R and the two distinct vertices are adjacent if and only if their product belongs to that ideal and at least one of them belongs to the sum of that ideal and the nilradical of the ring. The aim of this article is to study the structural properties of the N-prime ideal graph and establish the connection between the algebraic properties of ideals and their graph theoretic properties.In particular, the N-prime ideal graphs are characterized as star, pineapple, and complete multipartite graphs. The subgraphs induced by nilpotent and idempotent elements are investigated,along with their connections to zero-divisor and product maximal graphs. Furthermore, the properties of the N-prime ideal graph as a cograph and a threshold graph are studied and illustrated with examples. The study demonstrates how N-prime ideal properties can be effectively represented and analyzed using graph theoretic methods.

Published in	Applied and Computational Mathematics (Volume 15, Issue 6)
DOI	10.11648/j.acm.20261506.11
Page(s)	220-232
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

N-prime Ideal Graph, Primary Ideal, Nilradical, Nilpotent and Idempotent Elements, Complete Multipartite Graph, Cograph, Split Graph, Threshold Graph

References

[1]	D. S. Dummit and R. M. Foote, Abstract Algebra, 3rd ed. John Wiley & Sons, Hoboken, NJ, 2004. https://www.wiley.com/en-us/Abstract+Algebra%2C+3rd+Edition-p-9780471433347
[2]	J. A. Gallian, Contemporary Abstract Algebra, 9th ed. Cengage Learning, 2017. https://www.cengage.co.in/book-list/print/contemporary-abstract-algebra-9e-d6
[3]	D. B. West, Introduction to Graph Theory, 2nd ed. Prentice Hall, Upper Saddle River, NJ, 2001. https://dwest.web.illinois.edu/igt/
[4]	A. Cayley, "Desiderata and suggestions: No. 2. The theory of groups: graphical representation," American Journal of Mathematics, vol. 1, no. 2, pp. 174–176, 1878. https://doi.org/10.2307/2369306
[5]	J. Bosák, "The graphs of semigroups," in Theory of Graphs and Its Applications, M. Fiedler, Ed. Academic Press, New York, pp. 119–125, 1964. https://mathscinet.ams.org/mathscinet-getitem?mr=0173718
[6]	N. Biggs, Algebraic Graph Theory. Cambridge University Press, Cambridge, 1974. https://doi.org/10.1017/CBO9780511608704
[7]	I. Beck, "Coloring of commutative rings," Journal of Algebra, vol. 116, no. 1, pp. 208–226, 1988. https://doi.org/10.1016/0021-8693(88)90202-5
[8]	D. F. Anderson and P. S. Livingston, "The zero-divisor graph of a commutative ring," Journal of Algebra, vol. 217, no. 2, pp. 434–447, 1999. https://doi.org/10.1006/jabr.1998.7840
[9]	E. Şengelen Sevim and S. Koç, "On N-prime ideals," U.P.B. Scientific Bulletin, Series A, vol. 80, no. 2, pp. 177–188, 2018. https://www.scientificbulletin.upb.ro/rev_docs_arhiva/full2f0_971036.pdf
[10]	S. P. Redmond, "An ideal-based zero-divisor graph of a commutative ring," Communications in Algebra, vol. 31, no. 9, pp. 4425–4443, 2003. https://doi.org/10.1081/AGB-120022801
[11]	M. Behboodi and Z. Rakeei, "The annihilating-ideal graph of commutative rings I," Journal of Algebra and Its Applications, vol. 10, no. 4, pp. 727–739, 2011. https://doi.org/10.1142/S0219498811004896
[12]	A. Badawi, "On the annihilator graph of a commutative ring," Communications in Algebra, vol. 42, no. 1, pp. 108–121, 2014. https://doi.org/10.1080/00927872.2012.707262
[13]	M. J. Nikmehr and S. Khojasteh, "On the nilpotent graph of a ring," Turkish Journal of Mathematics, vol. 37, no. 4, pp. 553–559, 2013. https://doi.org/10.3906/mat-1112-35
[14]	M. J. Nikmehr, A. Azadi, and R. Nikandish, "The weakly zero-divisor graph of a commutative ring," Revista de la Unión Matemática Argentina, vol. 62, no. 1, pp. 105–116, 2021. https://doi.org/10.33044/revuma.1677
[15]	H. D. Patel, "The weakly annihilating-ideal graph of a commutative ring," Gulf Journal of Mathematics, vol. 14, no. 2, pp. 30–39, 2023. https://doi.org/10.56947/gjom.v14i2.1176
[16]	D. Kalamani and G. Ramya, "Product maximal graph of a finite commutative ring," Bulletin of the Calcutta Mathematical Society, vol. 113, no. 2, pp. 127–134, 2021. https://b-u.ac.in/users/kalamani
[17]	A. Hamed and A. Malek, "S-prime ideals of a commutative ring," Beiträge zur Algebra und Geometrie, vol. 61, no. 3, pp. 533–542, 2020. https://doi.org/10.1007/s13366-019-00476-5
[18]	F. A. A. Almahdi, E. M. Bouba, and M. Tamekkante, "On weakly S-prime ideals of commutative rings," Analele cStiinţifice ale Universităţii Ovidius Constanţa—Seria Matematică, vol. 29, no. 2, pp. 173–186, 2021. https://doi.org/10.2478/auom-2021-0024
[19]	D. Kalamani and C. V. Mythily, "S-prime ideal graph of a finite commutative ring," Advances and Applications in Mathematical Sciences, vol. 22, no. 4, pp. 861–872, 2023. https://www.mililink.com/issue_content.php?iId=409&id=59&is=4&mon=February&pg=809-882&vol=22&yer=2023
[20]	K. Duraisamy and V. Shanmugam, "On weakly S-prime ideal graph of a finite commutative ring," Applied and Computational Mathematics, vol. 15, no. 2, pp. 60–67, 2026. https://doi.org/10.11648/j.acm.20261502.12
[21]	H. Topcu, S. Sorgun, and W. H. Haemers, "On the spectral characterization of pineapple graphs," Linear Algebra and its Applications, vol. 507, pp. 267–273, 2016. https://doi.org/10.1016/j.laa.2016.06.018
[22]	V. Chvátal and P. L. Hammer, "Aggregation of inequalities in integer programming," Annals of Discrete Mathematics, vol. 1, pp. 145–162, 1977. https://doi.org/10.1016/S0167-5060(08)70731-3
[23]	D. Seinsche, "On a property of the class of n-colorable graphs," Journal of Combinatorial Theory, Series B, vol. 16, no. 2, pp. 191–193, 1974. https://doi.org/10.1016/0095-8956(74)90063-X
[24]	S. N. Daoud and A. N. Elsawy, "Edge even graceful labelling of some book graphs," Journal of Taibah University for Science, vol. 12, no. 3, pp. 315–330, 2018. https://doi.org/10.1080/16583655.2018.1469292
[25]	S. Főldes and P. L. Hammer, "Split graphs," in Proceedings of the Eighth Southeastern Conference on Combinatorics, Graph Theory and Computing, Congressus Numerantium, vol. 19, pp. 311–315, 1977.
[26]	P. L. Hammer and B. Simeone, "The splittance of a graph," Combinatorica, vol. 1, no. 3, pp. 275–284, 1981. https://doi.org/10.1007/BF02579333
[27]	X.-D. Hou, "Finite commutative chain rings," Finite Fields and Their Applications, vol. 7, no. 3, pp. 382–396, 2001. https://doi.org/10.1006/ffta.2000.0317
[28]	H. R. Maimani, M. R. Pournaki, and S. Yassemi, "Zero-divisor graph with respect to an ideal," Communications in Algebra, vol. 34, no. 3, pp. 923–929, 2006. https://doi.org/10.1080/00927870500441858
[29]	J. A. Bondy and U. S. R. Murty, Graph Theory, Graduate Texts in Mathematics, vol. 244, Springer, London, 2008. https://doi.org/10.1007/978-1-84628-970-5
[30]	S. N. Daoud and A. N. Elsawy, "Edge even graceful labelling of new families of graphs," Journal of Taibah University for Science, vol. 13, no. 1, pp. 579–591, 2019. https://doi.org/10.1080/16583655.2019.1611180
[31]	M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, Reading, Massachusetts, 1969. https://catalogue.bnf.fr/ark:/12148/cb373622871
[32]	S. H. Ahmad, M. Th. Al-Neima, Z. M. Ibraheem, and A. M. Ali, "Graph of orthogonal idempotent elements in rings of integers modulo n with spectral clustering," Journal of Mathematics, vol. 2025, Article ID 6853846, 13 pages, 2025. https://doi.org/10.1155/jom/6853846
[33]	D. F. Anderson and E. F. Lewis, A general theory of zero-divisor graphs over a commutative ring, International Electronic Journal of Algebra 20 (2016), 111–135. https://www.ieja.net/files/papers/volume-20/7-V20-2016.pdf

Cite This Article

Plain Text BibTeX RIS

APA Style

Duraisamy, K., Murugan, H. (2026). N-prime Ideal Graph of a Finite Commutative Ring. Applied and Computational Mathematics, 15(6), 220-232. https://doi.org/10.11648/j.acm.20261506.11

Copy | Download

ACS Style

Duraisamy, K.; Murugan, H. N-prime Ideal Graph of a Finite Commutative Ring. Appl. Comput. Math. 2026, 15(6), 220-232. doi: 10.11648/j.acm.20261506.11

Copy | Download

AMA Style

Duraisamy K, Murugan H. N-prime Ideal Graph of a Finite Commutative Ring. Appl Comput Math. 2026;15(6):220-232. doi: 10.11648/j.acm.20261506.11

Copy | Download

@article{10.11648/j.acm.20261506.11,
  author = {Kalamani Duraisamy and Hariharan Murugan},
  title = {N-prime Ideal Graph of a Finite Commutative Ring},
  journal = {Applied and Computational Mathematics},
  volume = {15},
  number = {6},
  pages = {220-232},
  doi = {10.11648/j.acm.20261506.11},
  url = {https://doi.org/10.11648/j.acm.20261506.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20261506.11},
  abstract = {In this paper, we introduce and investigate the N-prime ideal graph of a finite commutative ring with unity. The concept of prime ideal graphs over rings has been studied by many authors. Let R be a finite commutative ring with unity and let J be a proper ideal of R. The N-prime ideal graph of R with respect to J, denoted by GJ, is the simple undirected graph whose vertex set consists of the elements of the ring R and the two distinct vertices are adjacent if and only if their product belongs to that ideal and at least one of them belongs to the sum of that ideal and the nilradical of the ring. The aim of this article is to study the structural properties of the N-prime ideal graph and establish the connection between the algebraic properties of ideals and their graph theoretic properties.In particular, the N-prime ideal graphs are characterized as star, pineapple, and complete multipartite graphs. The subgraphs induced by nilpotent and idempotent elements are investigated,along with their connections to zero-divisor and product maximal graphs. Furthermore, the properties of the N-prime ideal graph as a cograph and a threshold graph are studied and illustrated with examples. The study demonstrates how N-prime ideal properties can be effectively represented and analyzed using graph theoretic methods.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - N-prime Ideal Graph of a Finite Commutative Ring
AU  - Kalamani Duraisamy
AU  - Hariharan Murugan
Y1  - 2026/11/05
PY  - 2026
N1  - https://doi.org/10.11648/j.acm.20261506.11
DO  - 10.11648/j.acm.20261506.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 220
EP  - 232
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20261506.11
AB  - In this paper, we introduce and investigate the N-prime ideal graph of a finite commutative ring with unity. The concept of prime ideal graphs over rings has been studied by many authors. Let R be a finite commutative ring with unity and let J be a proper ideal of R. The N-prime ideal graph of R with respect to J, denoted by GJ, is the simple undirected graph whose vertex set consists of the elements of the ring R and the two distinct vertices are adjacent if and only if their product belongs to that ideal and at least one of them belongs to the sum of that ideal and the nilradical of the ring. The aim of this article is to study the structural properties of the N-prime ideal graph and establish the connection between the algebraic properties of ideals and their graph theoretic properties.In particular, the N-prime ideal graphs are characterized as star, pineapple, and complete multipartite graphs. The subgraphs induced by nilpotent and idempotent elements are investigated,along with their connections to zero-divisor and product maximal graphs. Furthermore, the properties of the N-prime ideal graph as a cograph and a threshold graph are studied and illustrated with examples. The study demonstrates how N-prime ideal properties can be effectively represented and analyzed using graph theoretic methods.
VL  - 15
IS  - 6
ER  -

Copy | Download

Author Information

Kalamani Duraisamy

Department of Mathematics, Bharathiar University PG Extension and Research Centre, Erode, India

Contact Email

http://orcid.org/0009-0007-7974-5265
Hariharan Murugan

Department of Mathematics, Bharathiar University PG Extension and Research Centre, Erode, India

Contact Email

http://orcid.org/0009-0002-3544-443X

Download PDF

Submit an Article

Sections

Plain Text BibTeX RIS

APA Style

Duraisamy, K., Murugan, H. (2026). N-prime Ideal Graph of a Finite Commutative Ring. Applied and Computational Mathematics, 15(6), 220-232. https://doi.org/10.11648/j.acm.20261506.11

Copy | Download

ACS Style

Duraisamy, K.; Murugan, H. N-prime Ideal Graph of a Finite Commutative Ring. Appl. Comput. Math. 2026, 15(6), 220-232. doi: 10.11648/j.acm.20261506.11

Copy | Download

AMA Style

Duraisamy K, Murugan H. N-prime Ideal Graph of a Finite Commutative Ring. Appl Comput Math. 2026;15(6):220-232. doi: 10.11648/j.acm.20261506.11

Copy | Download

@article{10.11648/j.acm.20261506.11,
  author = {Kalamani Duraisamy and Hariharan Murugan},
  title = {N-prime Ideal Graph of a Finite Commutative Ring},
  journal = {Applied and Computational Mathematics},
  volume = {15},
  number = {6},
  pages = {220-232},
  doi = {10.11648/j.acm.20261506.11},
  url = {https://doi.org/10.11648/j.acm.20261506.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20261506.11},
  abstract = {In this paper, we introduce and investigate the N-prime ideal graph of a finite commutative ring with unity. The concept of prime ideal graphs over rings has been studied by many authors. Let R be a finite commutative ring with unity and let J be a proper ideal of R. The N-prime ideal graph of R with respect to J, denoted by GJ, is the simple undirected graph whose vertex set consists of the elements of the ring R and the two distinct vertices are adjacent if and only if their product belongs to that ideal and at least one of them belongs to the sum of that ideal and the nilradical of the ring. The aim of this article is to study the structural properties of the N-prime ideal graph and establish the connection between the algebraic properties of ideals and their graph theoretic properties.In particular, the N-prime ideal graphs are characterized as star, pineapple, and complete multipartite graphs. The subgraphs induced by nilpotent and idempotent elements are investigated,along with their connections to zero-divisor and product maximal graphs. Furthermore, the properties of the N-prime ideal graph as a cograph and a threshold graph are studied and illustrated with examples. The study demonstrates how N-prime ideal properties can be effectively represented and analyzed using graph theoretic methods.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - N-prime Ideal Graph of a Finite Commutative Ring
AU  - Kalamani Duraisamy
AU  - Hariharan Murugan
Y1  - 2026/11/05
PY  - 2026
N1  - https://doi.org/10.11648/j.acm.20261506.11
DO  - 10.11648/j.acm.20261506.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 220
EP  - 232
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20261506.11
AB  - In this paper, we introduce and investigate the N-prime ideal graph of a finite commutative ring with unity. The concept of prime ideal graphs over rings has been studied by many authors. Let R be a finite commutative ring with unity and let J be a proper ideal of R. The N-prime ideal graph of R with respect to J, denoted by GJ, is the simple undirected graph whose vertex set consists of the elements of the ring R and the two distinct vertices are adjacent if and only if their product belongs to that ideal and at least one of them belongs to the sum of that ideal and the nilradical of the ring. The aim of this article is to study the structural properties of the N-prime ideal graph and establish the connection between the algebraic properties of ideals and their graph theoretic properties.In particular, the N-prime ideal graphs are characterized as star, pineapple, and complete multipartite graphs. The subgraphs induced by nilpotent and idempotent elements are investigated,along with their connections to zero-divisor and product maximal graphs. Furthermore, the properties of the N-prime ideal graph as a cograph and a threshold graph are studied and illustrated with examples. The study demonstrates how N-prime ideal properties can be effectively represented and analyzed using graph theoretic methods.
VL  - 15
IS  - 6
ER  -

Copy | Download