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

TL;DR

This paper introduces the concept of $rak{X}$-elements in multiplicative lattices, generalizing classical ideals in rings, and establishes their properties and equivalences with known ideal types.

Contribution

It defines $rak{X}$-elements in multiplicative lattices, unifies various ideal concepts, and proves their correspondence with classical ring ideals.

Findings

$rak{X}$-elements generalize $r$-, $n$-, and $J$-ideals in rings.

Characterization of ideals as $rak{X}$-elements in lattice structures.

Equivalence between classical ideals and $rak{X}$-elements in ideal lattices.

Abstract

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 $R$.