Let ℋ be a finite commutative ring with unity. Let ℐ be a proper ideal of ℋ and 𝒮 is the multiplicative closed subset of ℋ which is disjoint with ℐ. The weakly S-prime ideal graph denoted by Gℐ(ℋ) is the undirected graph whose vertex set is the set of elements 𝔢 of ℋ such that the non-zero product ef is in ℐ and either se is in ℐ or sf is in ℐ for some f in ℋ and the two distinct vertices 𝔢 and 𝔣 are connected by an edge if and only if either se is in ℐ or sf is in ℐ for some s in 𝒮. The purpose of this article is to investigate the graph theoretic properties of the weakly S-prime ideal graph associated with ℋ. This study focuses on rings of order 2𝔭, 3𝔭 and 𝔭𝔮, where 𝔭 and 𝔮 are distinct primes. For these rings, the weakly S-prime ideal graph is a special type of graph and it is explained with examples. Furthermore, the graph theoretic concept of the weakly S-prime ideal graph Gℐ(ℋ) namely its girth, diameter, radius and size are studied. The relation between the weakly S-prime ideal graph and annihilator ideal graph associated with a ring of order 2𝔭 is described and it is proved that these two graphs are isomorphic.
| Published in | Applied and Computational Mathematics (Volume 15, Issue 2) |
| DOI | 10.11648/j.acm.20261502.12 |
| Page(s) | 60-67 |
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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. |
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Copyright © The Author(s), 2026. Published by Science Publishing Group |
Weakly S-prime Ideal, Degree, Diameter, Girth, Chromatic Number, Size
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APA Style
Duraisamy, K., Shanmugam, V. (2026). On Weakly S-Prime Ideal Graph Of a Finite Commutative Ring. Applied and Computational Mathematics, 15(2), 60-67. https://doi.org/10.11648/j.acm.20261502.12
ACS Style
Duraisamy, K.; Shanmugam, V. On Weakly S-Prime Ideal Graph Of a Finite Commutative Ring. Appl. Comput. Math. 2026, 15(2), 60-67. doi: 10.11648/j.acm.20261502.12
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Download@article{10.11648/j.acm.20261502.12,
author = {Kalamani Duraisamy and Vasuki Shanmugam},
title = {On Weakly S-Prime Ideal Graph Of a Finite Commutative Ring},
journal = {Applied and Computational Mathematics},
volume = {15},
number = {2},
pages = {60-67},
doi = {10.11648/j.acm.20261502.12},
url = {https://doi.org/10.11648/j.acm.20261502.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20261502.12},
abstract = {Let ℋ be a finite commutative ring with unity. Let ℐ be a proper ideal of ℋ and 𝒮 is the multiplicative closed subset of ℋ which is disjoint with ℐ. The weakly S-prime ideal graph denoted by Gℐ(ℋ) is the undirected graph whose vertex set is the set of elements 𝔢 of ℋ such that the non-zero product ef is in ℐ and either se is in ℐ or sf is in ℐ for some f in ℋ and the two distinct vertices 𝔢 and 𝔣 are connected by an edge if and only if either se is in ℐ or sf is in ℐ for some s in 𝒮. The purpose of this article is to investigate the graph theoretic properties of the weakly S-prime ideal graph associated with ℋ. This study focuses on rings of order 2𝔭, 3𝔭 and 𝔭𝔮, where 𝔭 and 𝔮 are distinct primes. For these rings, the weakly S-prime ideal graph is a special type of graph and it is explained with examples. Furthermore, the graph theoretic concept of the weakly S-prime ideal graph Gℐ(ℋ) namely its girth, diameter, radius and size are studied. The relation between the weakly S-prime ideal graph and annihilator ideal graph associated with a ring of order 2𝔭 is described and it is proved that these two graphs are isomorphic.},
year = {2026}
}
TY - JOUR T1 - On Weakly S-Prime Ideal Graph Of a Finite Commutative Ring AU - Kalamani Duraisamy AU - Vasuki Shanmugam Y1 - 2026/04/24 PY - 2026 N1 - https://doi.org/10.11648/j.acm.20261502.12 DO - 10.11648/j.acm.20261502.12 T2 - Applied and Computational Mathematics JF - Applied and Computational Mathematics JO - Applied and Computational Mathematics SP - 60 EP - 67 PB - Science Publishing Group SN - 2328-5613 UR - https://doi.org/10.11648/j.acm.20261502.12 AB - Let ℋ be a finite commutative ring with unity. Let ℐ be a proper ideal of ℋ and 𝒮 is the multiplicative closed subset of ℋ which is disjoint with ℐ. The weakly S-prime ideal graph denoted by Gℐ(ℋ) is the undirected graph whose vertex set is the set of elements 𝔢 of ℋ such that the non-zero product ef is in ℐ and either se is in ℐ or sf is in ℐ for some f in ℋ and the two distinct vertices 𝔢 and 𝔣 are connected by an edge if and only if either se is in ℐ or sf is in ℐ for some s in 𝒮. The purpose of this article is to investigate the graph theoretic properties of the weakly S-prime ideal graph associated with ℋ. This study focuses on rings of order 2𝔭, 3𝔭 and 𝔭𝔮, where 𝔭 and 𝔮 are distinct primes. For these rings, the weakly S-prime ideal graph is a special type of graph and it is explained with examples. Furthermore, the graph theoretic concept of the weakly S-prime ideal graph Gℐ(ℋ) namely its girth, diameter, radius and size are studied. The relation between the weakly S-prime ideal graph and annihilator ideal graph associated with a ring of order 2𝔭 is described and it is proved that these two graphs are isomorphic. VL - 15 IS - 2 ER -
Department of Mathematics, Bharathiar University PG Extension and Research Centre, Erode, India
Department of Mathematics, Bharathiar University PG Extension and Research Centre, Erode, India
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