Semiessential Fuzzy Ideals and Semiuniform Fuzzy Rings

Abstract

        Zadah in [1] introduced the notion of a fuzzy subset A of a nonempty set S as a mapping from S into [0,1], Liu in [2] introduced the concept of a fuzzy ring, Martines [3] introduced the notion of a fuzzy ideal of a fuzzy ring.         A non zero proper ideal I of a ring R is called an essential ideal if I  J  (0), for any non zero ideal J of R, [4].         Inaam in [5] fuzzified this concept to essential fuzzy ideal of fuzzy ring and gave its basic properties.         Nada in [6] introduced and studied notion of semiessential ideal in a ring R, where a non zero ideal I of R is called semiessential if I  P  (0) for all non zero prime ideals of R, [4].         A ring R is called uniform if every ideal of R is essential. Nada in [6] introduced and studied the notion semiuniform ring where a ring R is called semiuniform ring if every ideal of R is semiessential ideal.         In this paper we fuzzify the concepts semiessential ideal of a ring, uniform ring and semiuniform ring into semiessential fuzzy ideal of fuzzy ring, uniform fuzzy ring and semiuniform fuzzy ring. Where a fuzzy ideal A of a fuzzy ring X is semiessential if I  P  (0) for any prime fuzzy ideal P of X.         A fuzzy ring X is called uniform (semiuniform) if every fuzzy ideal of X is essential (semiessential) respectively.         In S.1, some basic definitions and results are collected.         In S.2, we study semiesential fuzzy ideals of fuzzy ring, we give some basic properties about this concept.         In S.3, we study the notion of uniform fuzzy rings and semiuniform fuzzy rings. Several properties about them are given.         Throughout this paper, R is commutative ring with unity, and X(0) = 1, for any fuzzy ring.