arXiv subjects

Sachin Sarode

Publications and source records attributed to Sachin Sarode.

6 recordsLinked to original sources

arXiv2026

On $S$-prime and $S$-primary elements in multiplicative lattices

In this paper, we study $S$-prime elements and $S$-primary elements within the framework of multiplicative lattices. Furthermore, we define and explore weakly $S$-prime elements and weakly $S$-primary elements, which generalize weakly prime elements and weakly primary elements in multiplicative lattices respectively. We show that the weakly $S$-prime ideals (weakly $S$-primary ideals) of a commutative ring $R$ with $1$ correspond precisely to the weakly $S_L$-prime elements (weakly $S$-primary elements) of the ideal lattice $Id(R)$ of $R$, where $S_L = \{(s) \mid s \in S\}$.

math.AC↗

arXiv2026

On $S$-Noetherian Lattices

In this paper, we define and study $S$-Noetherian lattices as a natural generalization of Noetherian rings. We prove that a ring $R$ is $S$-Noetherian if and only if its ideal lattice, $Id(R)$, is $S_L$-Noetherian. Furthermore, we establish a Cohen-Kaplansky type theorem for $S$-Noetherian lattices, showing that $L$ is $S$-Noetherian if and only if every $S$-prime element of $L$ is $S$-compact. Finally, we introduce the concept of $S$-primary elements-a generalization of primary elements in multiplicative lattices and demonstrate the existence and uniqueness of $S$-primary decomposition in $S$-Noetherian lattices.

math.AC↗

arXiv2026

On $S$-Prime Element Principle

In this paper, we introduce $S$-prime elements in $V$-lattices, where $S$ is a multiplicatively closed subset of a $V$-lattice $L$. In addition, we introduce the $S$-Prime Element Principle to prove that certain elements in $V$-lattices are $S$-prime elements. This principle leads to a direct and uniform approach to the results on the existence of prime elements in multiplicative lattices when $S=\{1\}$.

math.AC↗

arXiv2021

$\mathfrak{X}$-elements in multiplicative lattices -- A generalization of $J$-ideals, $n$-ideals and $r$-ideals in rings

In this paper, we introduce a concept of $\mathfrak{X}$-element with respect to an $M$-closed set $\mathfrak{X}$ in multiplicative lattices and study properties of $\mathfrak{X}$-elements. For a particular $M$-closed subset $\mathfrak{X}$, we define the concept of $r$-element, $n$-element and $J$-element. These elements generalize the notion of $r$-ideals, $n$-ideals and $J$-ideals of a commutative ring with unity to multiplicative lattices. In fact, we prove that an ideal $I$ of a commutative ring $R$ with unity is a $n$-ideal ($J$-ideal) of $R$ if and only if it is an $n$-element ($J$-element) of $Id(R)$, the ideal lattice of $R$.

math.AC↗

arXiv2013

Beck's Conjecture For Multiplicative Lattices

In this paper, we introduce the zero divisor graph of a multiplicative lattice. We provide a counter example to Beck's conjecture for multiplicative lattices. Further, we prove that Beck's conjecture is true for reduced multiplicative lattice which extends the result of Behboodi and Rakeei[7], and Aalipour et. al.[1].

math.AC↗

arXiv2013

Diameter and Girth of Zero Divisor Graph of Multiplicative Lattices

In this paper, we study the zero divisor graph $Γ^m(L)$ of a multiplicative lattice L. We prove under certain conditions that for a reduced multiplicative lattice L having more than two minimal prime elements, $Γ^m(L)$ contains a cycle and $gr(Γ^m(L)) = 3$. This essentially proves that for a reduced ring R with more than two minimal primes, $gr(\mathbb{AG}(R))) = 3$ which settles the conjecture of Behboodi and Rakeei [9]. Further, we have characterized the diameter of $Γ^m(L)$.

math.AC↗