Abstract

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

Keywords: Fuzzy ideal; prime fuzzy ideal; primary fuzzy ideal; (strongly) irreducible fuzzy ideal; (strongly) Laskerian ring.

MSC: 13A15, 03E72, 13C12

Pages:  74--81     

Volume  65 ,  Issue  1 ,  2013