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

In this paper, we introduce a concept of X-element with respect to an M -closed set X in multiplicative lattices and study properties of X-elements. For a particular M -closed subset 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.