Spectrum of prime fuzzy ideals

Abstract

A topological space called a spectrum of prime fuzzy ideals and denoted by Fspec( R) is obtained by introducing a topology on the set of prime fuzzy ideals of a ring R. The prime fuzzy ideals used here (introduced by the author elsewhere), are not necessarily two valued, the topology introduced is compact and the usual (nonfuzzy) prime spectrum Spec( R) is a subspace of Fspec( R) and dense in it. Situations under which fuzzy spectrums of different rings are homeomorphic are discussed. In particular, Fspec( R) is proved to be homeomorphic to Fspec( R/Frad( R)), where Frad( R) is the fuzzy prime radical of R. The correspondence associating a ring R to the topological space Fspec( R) is shown to define a contravarient functor from the category of rings with unity, into the category of compact topological spaces.