  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 $R$.
