Let mathfrakB be a finite commutative ring with unity. Let I be the k-prime ideal of mathfrakB which is a generalization of prime ideal. For any fixed integer kgeq 3, an ideal I of a ring mathfrakB is said to be k-prime whenever for any set G of nonzero, distinct, and non-unit elements of mathfrakB, the product g1.g2.g3...gk belongs to I implies that the product of the elements of a proper subset of G is in I. This paper investigates the algebraic structure of k-prime ideal in finite commutative ring Zn and studies their behaviour under ideal operations. Initially it is proved that every prime ideal of a ring mathfrakB is k-prime and also every non-zero ideal is kM-prime. The behaviour of k-prime ideal under sum, product and union is then examined. Also, some ring theoretic properties related to k-prime ideals are discussed with suitable examples. Furthermore, it is proved that if I is a k-prime ideal of mathfrakB, then rad(I) is a k-prime ideal of mathfrakB. Moreover the number of radical ideals in a finite commutative ring mathfrakB is generalized. The study is further extended to minimal prime ideals over a given ideal and their relationship with k-prime ideals. A generalized formula for the length of the ideal is established. Finally, the relationship among k-prime ideals, minimal primes over an ideal and length of the ideal are illustrated through suitable examples.
| Published in | Applied and Computational Mathematics (Volume 15, Issue 5) |
| DOI | 10.11648/j.acm.20261505.13 |
| Page(s) | 178-184 |
| 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 |
Zero-divisor, k-zero-divisor, Radical Ideal, Minimal Primes Over I, Chain
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APA Style
Duraisamy, K., Sharavanan, J. A. (2026). On k-prime Ideal of a Finite Commutative Ring. Applied and Computational Mathematics, 15(5), 178-184. https://doi.org/10.11648/j.acm.20261505.13
ACS Style
Duraisamy, K.; Sharavanan, J. A. On k-prime Ideal of a Finite Commutative Ring. Appl. Comput. Math. 2026, 15(5), 178-184. doi: 10.11648/j.acm.20261505.13
AMA Style
Duraisamy K, Sharavanan JA. On k-prime Ideal of a Finite Commutative Ring. Appl Comput Math. 2026;15(5):178-184. doi: 10.11648/j.acm.20261505.13
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Download@article{10.11648/j.acm.20261505.13,
author = {Kalamani Duraisamy and Jayasree Anbuselvi Sharavanan},
title = {On k-prime Ideal of a Finite Commutative Ring},
journal = {Applied and Computational Mathematics},
volume = {15},
number = {5},
pages = {178-184},
doi = {10.11648/j.acm.20261505.13},
url = {https://doi.org/10.11648/j.acm.20261505.13},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20261505.13},
abstract = {Let mathfrakB be a finite commutative ring with unity. Let I be the k-prime ideal of mathfrakB which is a generalization of prime ideal. For any fixed integer kgeq 3, an ideal I of a ring mathfrakB is said to be k-prime whenever for any set G of nonzero, distinct, and non-unit elements of mathfrakB, the product g1.g2.g3...gk belongs to I implies that the product of the elements of a proper subset of G is in I. This paper investigates the algebraic structure of k-prime ideal in finite commutative ring Zn and studies their behaviour under ideal operations. Initially it is proved that every prime ideal of a ring mathfrakB is k-prime and also every non-zero ideal is kM-prime. The behaviour of k-prime ideal under sum, product and union is then examined. Also, some ring theoretic properties related to k-prime ideals are discussed with suitable examples. Furthermore, it is proved that if I is a k-prime ideal of mathfrakB, then rad(I) is a k-prime ideal of mathfrakB. Moreover the number of radical ideals in a finite commutative ring mathfrakB is generalized. The study is further extended to minimal prime ideals over a given ideal and their relationship with k-prime ideals. A generalized formula for the length of the ideal is established. Finally, the relationship among k-prime ideals, minimal primes over an ideal and length of the ideal are illustrated through suitable examples.},
year = {2026}
}
TY - JOUR T1 - On k-prime Ideal of a Finite Commutative Ring AU - Kalamani Duraisamy AU - Jayasree Anbuselvi Sharavanan Y1 - 2026/09/24 PY - 2026 N1 - https://doi.org/10.11648/j.acm.20261505.13 DO - 10.11648/j.acm.20261505.13 T2 - Applied and Computational Mathematics JF - Applied and Computational Mathematics JO - Applied and Computational Mathematics SP - 178 EP - 184 PB - Science Publishing Group SN - 2328-5613 UR - https://doi.org/10.11648/j.acm.20261505.13 AB - Let mathfrakB be a finite commutative ring with unity. Let I be the k-prime ideal of mathfrakB which is a generalization of prime ideal. For any fixed integer kgeq 3, an ideal I of a ring mathfrakB is said to be k-prime whenever for any set G of nonzero, distinct, and non-unit elements of mathfrakB, the product g1.g2.g3...gk belongs to I implies that the product of the elements of a proper subset of G is in I. This paper investigates the algebraic structure of k-prime ideal in finite commutative ring Zn and studies their behaviour under ideal operations. Initially it is proved that every prime ideal of a ring mathfrakB is k-prime and also every non-zero ideal is kM-prime. The behaviour of k-prime ideal under sum, product and union is then examined. Also, some ring theoretic properties related to k-prime ideals are discussed with suitable examples. Furthermore, it is proved that if I is a k-prime ideal of mathfrakB, then rad(I) is a k-prime ideal of mathfrakB. Moreover the number of radical ideals in a finite commutative ring mathfrakB is generalized. The study is further extended to minimal prime ideals over a given ideal and their relationship with k-prime ideals. A generalized formula for the length of the ideal is established. Finally, the relationship among k-prime ideals, minimal primes over an ideal and length of the ideal are illustrated through suitable examples. VL - 15 IS - 5 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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