X\mathfrak{X}X-elements in multiplicative lattices - A generalization of JJJ-ideals, nnn-ideals and rrr-ideals in rings (original) (raw)
International Electronic Journal of Algebra
In this paper, we introduce the concept of an mathfrakX\mathfrak{X}mathfrakX-element with respect to an MMM-closed set mathfrakX\mathfrak{X}mathfrakX in multiplicative lattices and study properties of mathfrakX\mathfrak{X}mathfrakX-elements. For a particular MMM-closed subset mathfrakX\mathfrak{X}mathfrakX, we define the concepts of rrr-elements, nnn-elements and JJJ-elements. These elements generalize the notion of rrr-ideals, nnn-ideals and JJJ-ideals of a commutative ring with identity to multiplicative lattices. In fact, we prove that an ideal III of a commutative ring RRR with identity is a nnn-ideal ($J$-ideal) of RRR if and only if it is an nnn-element ($J$-element) of Id(R)Id(R)Id(R), the ideal lattice of RRR.
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