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
DOI:
10.11648/j.acm.20261505.11
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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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