Fuzzy ideals in Laskerian rings

Abstract

We introduce strongly primary fuzzy ideals and strongly irreducible fuzzyideals in a unitary commutative ring and fixed their role in a Laskerianring. We established that: A finite intersection of prime fuzzy ideals(resp. primary fuzzy ideals, irreducible fuzzy ideals and stronglyirreducible fuzzy ideals) is a prime fuzzy ideal (resp. primary fuzzy ideal,irreducible fuzzy ideal and strongly irreducible fuzzy ideal). We also findthat, a fuzzy ideal of a ring is prime if and only if it is semiprime andstrongly irreducible. Furthermore we characterize that: (1) Every nonzerofuzzy ideal of a one dimensional Laskerian domain can be uniquely expressedas a product of primary fuzzy ideals with distinct radicals, (2) A unitarycommutative ring is (strongly) Laskerian if and only if its localization is(strongly) Laskerian with respect to every fuzzy ideal.