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

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 $R$.