A multiplicative lattice is a complete lattice endowed with a commutative, associative, join distributive multiplication in which 1 acts as a multiplicative identity, providing an abstract framework for the study of the ideal theory of commutative rings. In this paper, the concept of an S-prime element is introduced, generalizing the notion of an S-prime ideal of a commutative ring with unity to multiplicative lattices. Let L be a multiplicative lattice with k distinct prime elements and S be the multiplicative closed subset of L. Initially, the fundamental characterizations of S-prime elements are investigated. Furthermore, it is shown that every element of a reduced lattice is an idempotent and radical element, respectively. In particular, every proper element of a reduced lattice is an S-prime element. The number of S-prime elements in a given multiplicative is determined and S is the upset of k-1 prime elements in L. Moreover, the sets of non S-prime elements and nilpotent elements have the same cardinality in L. It is further shown that the radical of every nilpotent element is an S-prime element. The uniqueness of an S-prime element in a local lattice is proved and consequently, every zero divisor is a nilpotent element. Finally, several illustrative examples are presented to demonstrate the structural behavior of S-prime elements in L.
| Published in | Applied and Computational Mathematics (Volume 15, Issue 5) |
| DOI | 10.11648/j.acm.20261505.11 |
| Page(s) | 162-167 |
| 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 |
Multiplicative Lattice, Reduced Lattice, Prime Element, Nilpotent Element, Zero-divisor
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APA Style
Duraisamy, K., Arockiasamy, M. C. (2026). On S-prime Elements and their Generalizations in Multiplicative Lattices. Applied and Computational Mathematics, 15(5), 162-167. https://doi.org/10.11648/j.acm.20261505.11
ACS Style
Duraisamy, K.; Arockiasamy, M. C. On S-prime Elements and their Generalizations in Multiplicative Lattices. Appl. Comput. Math. 2026, 15(5), 162-167. doi: 10.11648/j.acm.20261505.11
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Download@article{10.11648/j.acm.20261505.11,
author = {Kalamani Duraisamy and Movis Chelcea Arockiasamy},
title = {On S-prime Elements and their Generalizations in Multiplicative Lattices},
journal = {Applied and Computational Mathematics},
volume = {15},
number = {5},
pages = {162-167},
doi = {10.11648/j.acm.20261505.11},
url = {https://doi.org/10.11648/j.acm.20261505.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20261505.11},
abstract = {A multiplicative lattice is a complete lattice endowed with a commutative, associative, join distributive multiplication in which 1 acts as a multiplicative identity, providing an abstract framework for the study of the ideal theory of commutative rings. In this paper, the concept of an S-prime element is introduced, generalizing the notion of an S-prime ideal of a commutative ring with unity to multiplicative lattices. Let L be a multiplicative lattice with k distinct prime elements and S be the multiplicative closed subset of L. Initially, the fundamental characterizations of S-prime elements are investigated. Furthermore, it is shown that every element of a reduced lattice is an idempotent and radical element, respectively. In particular, every proper element of a reduced lattice is an S-prime element. The number of S-prime elements in a given multiplicative is determined and S is the upset of k-1 prime elements in L. Moreover, the sets of non S-prime elements and nilpotent elements have the same cardinality in L. It is further shown that the radical of every nilpotent element is an S-prime element. The uniqueness of an S-prime element in a local lattice is proved and consequently, every zero divisor is a nilpotent element. Finally, several illustrative examples are presented to demonstrate the structural behavior of S-prime elements in L.},
year = {2026}
}
TY - JOUR T1 - On S-prime Elements and their Generalizations in Multiplicative Lattices AU - Kalamani Duraisamy AU - Movis Chelcea Arockiasamy Y1 - 2026/09/01 PY - 2026 N1 - https://doi.org/10.11648/j.acm.20261505.11 DO - 10.11648/j.acm.20261505.11 T2 - Applied and Computational Mathematics JF - Applied and Computational Mathematics JO - Applied and Computational Mathematics SP - 162 EP - 167 PB - Science Publishing Group SN - 2328-5613 UR - https://doi.org/10.11648/j.acm.20261505.11 AB - A multiplicative lattice is a complete lattice endowed with a commutative, associative, join distributive multiplication in which 1 acts as a multiplicative identity, providing an abstract framework for the study of the ideal theory of commutative rings. In this paper, the concept of an S-prime element is introduced, generalizing the notion of an S-prime ideal of a commutative ring with unity to multiplicative lattices. Let L be a multiplicative lattice with k distinct prime elements and S be the multiplicative closed subset of L. Initially, the fundamental characterizations of S-prime elements are investigated. Furthermore, it is shown that every element of a reduced lattice is an idempotent and radical element, respectively. In particular, every proper element of a reduced lattice is an S-prime element. The number of S-prime elements in a given multiplicative is determined and S is the upset of k-1 prime elements in L. Moreover, the sets of non S-prime elements and nilpotent elements have the same cardinality in L. It is further shown that the radical of every nilpotent element is an S-prime element. The uniqueness of an S-prime element in a local lattice is proved and consequently, every zero divisor is a nilpotent element. Finally, several illustrative examples are presented to demonstrate the structural behavior of S-prime elements in L. 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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