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Research Article
On S-prime Elements and their Generalizations in Multiplicative Lattices
Kalamani Duraisamy
,
Movis Chelcea Arockiasamy*
Issue:
Volume 15, Issue 5, October 2026
Pages:
162-167
Received:
17 July 2026
Accepted:
30 July 2026
Published:
1 September 2026
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.
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...
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Research Article
A Study on the Chromatic Numbers of Fuzzy Graphs
Sasirekha Rathinasamy,
Sathiya Palaniappan*
Issue:
Volume 15, Issue 5, October 2026
Pages:
168-177
Received:
23 July 2026
Accepted:
27 August 2026
Published:
18 September 2026
Abstract: Fuzzy graph coloring is an important area of fuzzy graph theory that generalizes the traditional concept of graph coloring to environments where uncertainty and ambiguity are inherent. By incorporating membership values into graph structures, fuzzy coloring provides an effective framework for representing imprecise relationships, making it valuable in numerous applications such as communication networks, transportation planning, scheduling, resource management, pattern recognition, and decision-making systems. Let G=(V, σ, µ) be a fuzzy graph. A fuzzy coloring of G is an assignment of either basic colors or fuzzy colors to the vertices while satisfying the coloring conditions determined by the strengths of the connecting edges. A coloring is regarded as proper if any two adjacent vertices connected by a strong edge are assigned distinct basic colors or distinct fuzzy colors whenever required. Alternatively, one vertex may receive a basic color and the other a fuzzy color associated with a different basic color. On the other hand, when two adjacent vertices are linked by a weak edge, they may be assigned identical fuzzy colors, different fuzzy colors, or a combination in which one vertex is given a basic color and the other a fuzzy color corresponding to the same basic color. The minimum number of basic and fuzzy colors required to obtain a proper coloring is called the fuzzy chromatic number, denoted by χf (G). This study develops an enhanced fuzzy coloring approach and applies it to determine the fuzzy chromatic numbers of several families of fuzzy graphs, including fuzzy helm graphs, fuzzy trees, and fuzzy caterpillar graphs. Rigorous mathematical analysis is employed to establish the corresponding results. Furthermore, an application is presented to demonstrate the usefulness of fuzzy coloring and the fuzzy chromatic number as effective tools for modelling and analyzing real-world systems characterized by uncertain or imprecise relationships.
Abstract: Fuzzy graph coloring is an important area of fuzzy graph theory that generalizes the traditional concept of graph coloring to environments where uncertainty and ambiguity are inherent. By incorporating membership values into graph structures, fuzzy coloring provides an effective framework for representing imprecise relationships, making it valuable...
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Research Article
On k-prime Ideal of a Finite Commutative Ring
Kalamani Duraisamy,
Jayasree Anbuselvi Sharavanan*
Issue:
Volume 15, Issue 5, October 2026
Pages:
178-184
Received:
3 September 2026
Accepted:
3 September 2026
Published:
24 September 2026
DOI:
10.11648/j.acm.20261505.13
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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.
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 th...
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