Kumbhojkar, H.V. (Prof)Jemal, Saadu2022-04-122023-11-042022-04-122023-11-042005-06http://etd.aau.edu.et/handle/123456789/31277This paper has three parts: The first part deals with preliminaries on semi groups, spectrum of prime ideals of a semi group and preliminaries on fuzzy sets that are going to be used for the works in the following parts. In the second part primary fuzzy ideals, semi prime fuzzy ideals and nil and prime radical of a fuzzy ideal are discussed and their relationship with prime fuzzy ideals is investigated. It is also proved that the nil and prime ideals coincide when the grade membership lattice is totally ordered. In the last, a topological space called spectrum of fuzzy ideals, FSpec(S), is obtained by introducing a topology on the set of prime fuzzy ideals of a semi group S. It is also proved that FSpec(S) is compact and the non-fuzzy prime spectrum, Spec(S) , is dense in FSpec(S). It is shown that if S and S' are isomorphic, then FSpec(S) and FSpec(S') are homeomorphic. Key words: Fuzzy ideal, Prime fuzzy ideal, Semi prime fuzzy ideal, Primary fuzzy ideal, Radical of a fuzzy ideal, Compact topological space.enFuzzy idealPrime fuzzy idealSemi prime fuzzy idealPrimary fuzzy idealRadical of a fuzzy idealCompact topological spaceThesis