Fuzzy Sets and Systems 42 (1991) 263-272 263 North-Holland Fuzzy semiprimary ideals of rings* Rajesh Kumar Department of Mathematics, P. G. D.A. V. College (E), (University of Delhi), Nehru Nagar, Ring Road, New Delhi-110065, India Received April 1989 Abstract: The concept of fuzzy ideals is extended by introducing fuzzy semiprimary ideals in rings. This class of fuzzy ideals generalizes the class of fuzzy semiprimary ideals. It is shown that there do exist fuzzy ideals which are fuzzy semiprimary but not fuzzy primary. A study of (i) the level ideals of a fuzzy semiprimary ideal (ii) the characteristic function of a semiprimary ideal (iii) the zero divisors in the ring of all fuzzy cosets of a fuzzy semiprimary ideal and (iv) the algebraic nature of direct and inverse images of fuzzy semiprimary ideals under homomorphisms, is carried out. Keywords: Fuzzy prime ideal; fuzzy primary ideal; fuzzy semiprimary ideal; fuzzy cosets of a fuzzy ideal. 1. Introduction Zadeh, in his classic paper [14], introduced the notions of fuzzy sets and fuzzy set operations. Since then the fuzzy set theory developed by Zadeh and others has evoked great interest among researchers working in different branches of mathematics. Liu, in his pioneering paper [9], introduced and studied the idea of fuzzy subrings and fuzzy ideals. Subsequently, Mukherjee and Sen [10], Swamy and Swamy [12], Yue [13], Dixit et al. [4, 5] and Rajesh [7] applied some basic concepts pertaining to ideals from classical ring theory and developed a theory of fuzzy ideals. Fuzzy subrings (fuzzy ideals) are different from ordinary subrings (ideals) in the sense that it is not possible to decide which ring elements belong to the fuzzy subring (fuzzy ideal) under consideration and which do not. However, if/~ is any fuzzy subring (fuzzy ideal) of a ring R and x is any element of R, then the smallest level subring (level ideal) containing x always exists. Section 2 of this paper is a prerequisite for the rest of the paper. In this section, some basic definitions and results are discussed. In Section 3, a fuzzy semiprimary ideal is defined. This class of fuzzy ideals generalizes the class of fuzzy primary ideals considered by Yue [13] and Rajesh [7]. It is shown that an ideal of a commutative ring R with unity is semiprimary if and only if its characteristic function is a fuzzy semiprimary ideal of the given ring. This result enables us to extend the concept of fuzzy semiprimary ideals and motivates for further * Dedicated to Dr. V.N. Dixit and Dr. Naseem Ajmal (University of Delhi). 0165-0114/91/$03.50 © 1991--Elsevier Science Publishers B.V. (North-Holland)