When do L-fuzzy ideals of a ring generate a distributive lattice?

Abstract

Abstract:The notion of L-fuzzy extended ideals is introduced in a Boolean ring, and their essential properties are investigated. We also build the relation between an L-fuzzy ideal and the class of its L-fuzzy extended ideals. By defining an operator " " between two arbitrary L-fuzzy ideals in terms of L-fuzzy extended ideals, the result that "the family of all L-fuzzy ideals in a Boolean ring is a complete Heyting algebra" is immediately obtained. Furthermore, the lattice structures of L-fuzzy extended ideals of an L-fuzzy ideal, L-fuzzy extended ideals relative to an L-fuzzy subset, L-fuzzy stable ideals relative to an L-fuzzy subset and their connections are studied in this paper.

A note on distributivity of the lattice of L-ideals of a ring

“…Recently in [11] the modularity of L-ideals of a ring is established, where the subdirect product theorem of Tom Head does not apply. Finally, the lattice of L-fuzzy extended ideals is studied in [7]. We ask: is the lattice of all L-ideals of a ring distributive whose lattice of all ideals is distributive?…”