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

Authors : Sachin SARODE, Vinayak JOSHI

Pages : 46-61

Doi:10.24330/ieja.1102289

View : 16 | Download : 7

Publication Date : 2022-07-16

Article Type : Research Paper

Abstract :In this paper, we introduce the concept of an $\mathfrak{X}$-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 concepts of $r$-elements, $n$-elements and $J$-elements. These elements generalize the notion of $r$-ideals, $n$-ideals and $J$-ideals of a commutative ring with identity to multiplicative lattices. In fact, we prove that an ideal $I$ of a commutative ring $R$ with identity is a $n$-ideal insert ignore into journalissuearticles values($J$-ideal); of $R$ if and only if it is an $n$-element insert ignore into journalissuearticles values($J$-element); of $Idinsert ignore into journalissuearticles values(R);$, the ideal lattice of $R$.
Keywords : mathfrak X element, n element, r element, n ideal, r ideal, J ideal, J element, Multiplicative lattice, prime element, commutative ring