Let ℜ be the finite commutative ring with unity and Is be the S-prime ideal of a ring ℜ. The set Ls forms a partially ordered set (poset) by the subset relation. Initially, the interplay of the semilattice theoretic properties of a poset with the ring theoretic properties are studied with suitable examples. The number of maximal chain of a poset is compared with the number of prime ideals of a ring. It is proved that every maximal element of a poset is the prime ideal of a ring. A prime order ring is shown as a lattice. If the order of the ring is the product of two primes, then the trivial ideal is expressed as the meet of every pair of a poset. Further, the cardinality of the poset is determined in terms of the divisors of the order of the ring . A new meet-semilattice called the S-meet semilattice (Ls, ⋀, ⊆) is defined and the generalized Hasse diagrams of the S-meet semilattice of a ring of prime powers, product of prime powers are drawn in this paper in order to find the properties of S-meet semilattice. Finally, the ideals, the prime ideals and the maximal ideals of the S-meet semilattice are described in terms of the down-sets of S-meet semilattice where the results are listed with an example at the end.
| Published in | Applied and Computational Mathematics (Volume 13, Issue 4) |
| DOI | 10.11648/j.acm.20241304.14 |
| Page(s) | 105-110 |
| 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. |
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Copyright © The Author(s), 2024. Published by Science Publishing Group |
Prime Ideal, S-prime Ideal, Nilpotent Ideal, Meet-semilattice
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APA Style
Duraisamy, K., Varadharajan, M. (2024). Some New Results on S-prime Ideals of a Finite Commutative Ring as S-meet Semilattice. Applied and Computational Mathematics, 13(4), 105-110. https://doi.org/10.11648/j.acm.20241304.14
ACS Style
Duraisamy, K.; Varadharajan, M. Some New Results on S-prime Ideals of a Finite Commutative Ring as S-meet Semilattice. Appl. Comput. Math. 2024, 13(4), 105-110. doi: 10.11648/j.acm.20241304.14
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Download@article{10.11648/j.acm.20241304.14,
author = {Kalamani Duraisamy and Mythily Varadharajan},
title = {Some New Results on S-prime Ideals of a Finite Commutative Ring as S-meet Semilattice},
journal = {Applied and Computational Mathematics},
volume = {13},
number = {4},
pages = {105-110},
doi = {10.11648/j.acm.20241304.14},
url = {https://doi.org/10.11648/j.acm.20241304.14},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20241304.14},
abstract = {Let ℜ be the finite commutative ring with unity and Is be the S-prime ideal of a ring ℜ. The set Ls forms a partially ordered set (poset) by the subset relation. Initially, the interplay of the semilattice theoretic properties of a poset with the ring theoretic properties are studied with suitable examples. The number of maximal chain of a poset is compared with the number of prime ideals of a ring. It is proved that every maximal element of a poset is the prime ideal of a ring. A prime order ring is shown as a lattice. If the order of the ring is the product of two primes, then the trivial ideal is expressed as the meet of every pair of a poset. Further, the cardinality of the poset is determined in terms of the divisors of the order of the ring . A new meet-semilattice called the S-meet semilattice (Ls, ⋀, ⊆) is defined and the generalized Hasse diagrams of the S-meet semilattice of a ring of prime powers, product of prime powers are drawn in this paper in order to find the properties of S-meet semilattice. Finally, the ideals, the prime ideals and the maximal ideals of the S-meet semilattice are described in terms of the down-sets of S-meet semilattice where the results are listed with an example at the end.},
year = {2024}
}
TY - JOUR T1 - Some New Results on S-prime Ideals of a Finite Commutative Ring as S-meet Semilattice AU - Kalamani Duraisamy AU - Mythily Varadharajan Y1 - 2024/08/08 PY - 2024 N1 - https://doi.org/10.11648/j.acm.20241304.14 DO - 10.11648/j.acm.20241304.14 T2 - Applied and Computational Mathematics JF - Applied and Computational Mathematics JO - Applied and Computational Mathematics SP - 105 EP - 110 PB - Science Publishing Group SN - 2328-5613 UR - https://doi.org/10.11648/j.acm.20241304.14 AB - Let ℜ be the finite commutative ring with unity and Is be the S-prime ideal of a ring ℜ. The set Ls forms a partially ordered set (poset) by the subset relation. Initially, the interplay of the semilattice theoretic properties of a poset with the ring theoretic properties are studied with suitable examples. The number of maximal chain of a poset is compared with the number of prime ideals of a ring. It is proved that every maximal element of a poset is the prime ideal of a ring. A prime order ring is shown as a lattice. If the order of the ring is the product of two primes, then the trivial ideal is expressed as the meet of every pair of a poset. Further, the cardinality of the poset is determined in terms of the divisors of the order of the ring . A new meet-semilattice called the S-meet semilattice (Ls, ⋀, ⊆) is defined and the generalized Hasse diagrams of the S-meet semilattice of a ring of prime powers, product of prime powers are drawn in this paper in order to find the properties of S-meet semilattice. Finally, the ideals, the prime ideals and the maximal ideals of the S-meet semilattice are described in terms of the down-sets of S-meet semilattice where the results are listed with an example at the end. VL - 13 IS - 4 ER -
Department of Mathematics, Bharathiar University PG Extension and Research Centre, Erode, Tamil Nadu, India
Department of Mathematics, Bharathiar University PG Extension and Research Centre, Erode, Tamil Nadu, India
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