SOOCHOW JOURNAL OF MATHEMATICS Volume No pp October EFFECTIVE GENERATORS FOR FUZZY IDEALS BY

We give new sharp bounds for the number of generators for fuzzy ideals The pioneering works of Zadeh and Rosenfeld led to the fuzzi cation of algebraic structures Liu introduced and examined the notion of a fuzzy ideal of a ring Malik and Mordeson have been studying properties of fuzzy ideals see for example and The concept of minimal generating systems for fuzzy ideals was introduced in It has also been further investigated in In particular it is shown that for a commutative ring R of length n each fuzzy ideal has a system of fuzzy generators consisting of no more than n elements Theorem iii We improve this bound in this work and show with examples that the new bounds are sharp We start recalling some de nitions and results from and then we prove in the main Theorem that in a ring of length n a fuzzy ideal with jIm j m needs no more thanm n m generators Then we give new bounds in Corollaries and that improve the mentioned bound in Theorem iii We show that every fuzzy ideal has a system of generators with no more than n n elements Corollary and that for Artinian rings n n generators are enough Corollary Examples and show that these bounds are sharp Let R be be a commutative ring and let and be fuzzy subsets of R i e functions of R into We say that if and only if x x x R We say that if and only if and x R such that x x If is a fuzzy subset of R we let Supp denote fx Rj x g the support of We Received June AMS Subject Classi cation A L E