IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 17, Issue 2 Ser. II (Mar. – Apr. 2021), PP 44-50 www.iosrjournals.org DOI: 10.9790/5728-1702024450 www.iosrjournals.org 44 | Page Fuzzy Rings and Anti Fuzzy Rings With Operators M.Z.ALAM 1 & AMIT KUMAR ARYA 2 1. COLLEGE OF COMMERCE. ARTS & SCIENCE,PATNA (BIHAR) INDIA. 2. Research Scholar ,Dept. of Mathematics, M. U. Bodh Gaya (BIHAR) INDIA Abstract: In this paper, we studied the theory of fuzzy rings, the concept of fuzzy ring with operators, fuzzy ideal and anti-fuzzy ideal with operators, fuzzy homomorphism with operators etc., and their some elementary properties. Keywords: Fuzzy rings, fuzzy ring with operators, fuzzy ideal with operators, anti-fuzzy ideal, and homomorphism. --------------------------------------------------------------------------------------------------------------------------------------- Date of Submission: 18-03-2021 Date of Acceptance: 01-04-2021 --------------------------------------------------------------------------------------------------------------------------------------- I. Introduction: In 1982 W.J.Liu [1] introduced the concept of fuzzy ring. In 1985 Y.C.Ren [2] established the notion of fuzzy ideal and quotient ring. In this paper we studied the theory of fuzzy ring, and the concept of fuzzy ring with operators, fuzzy ideal and anti-fuzzy ideal with operators, homomorphism and their related elementary properties have discussed. II. Prliminaries Definition 2.1:[Liu [1]] Let R be a ring, a fuzzy set A of R is called a fuzzy ring of R if (i) A(x – y) ≥ min ( A(x), A(y) ), for all x, y in R (ii) A(x y) ≥ min (A(x), A(y) ), for all x, y in R Definition 2.2: [Liu [1]] Let R be a ring, a fuzzy ring A of R is called a ring with operator ( read as M – fuzzy ring ) iff for any t  [0,1],   is a ring with operator of R (i.e M – subring of R ), when   . Where           Definition 2.3: Let A be M – fuzzy ideal of R, is a M – fuzzy subring of R such that (iii) A(y + x – y)  A(x) (iv) A(x y)   (v) A((x + z)y – x y ) (z) For all x, y, z  R Note that A is a M – fuzzy left ideal of R if it satisfies (i), (ii), (iii) and (iv), and A is said to be a M – fuzzy right ideal of R, if it satisfies (i), (ii), (iii) and (v). Definition 2.4: Let R be a ring, a fuzzy set A of R is called anti fuzzy subring of R, if for all x, y,  (AF 1 ) A(x – y)     (AF 2 ) A(x y)     Definition 2.5: Let R be a ring, a fuzzy ring A of R is called an anti-fuzzy subring with operator ( read as anti M- fuzzy subring) iff for any t  ,   is an anti-ring with operator of R (i.e.anti M-fuzzy subring of R), when     Where            Definition 2.6: Let A be M-fuzzy anti ideal of R, if A is a anti M-fuzzy subring of R such that the following conditions are satisfied (AF 3 ) A(y + x – y)   (AF 4 ) A(x y)   (AF 5 ) A ((x + z) y - x y)   For all x, y, z . Note that A is an anti M-fuzzy left ideal of R if it satisfies (AF 1 ), (AF 2 ), (AF 3 ) and (AF 4 ), and A is called an anti M-fuzzy right ideal of R if it satisfies (AF 1 ), (AF 2 ), (AF 3 ) and (AF 5 ). Example: Let R = {         }, be a set with two binary operations as follows 1 Asso.Prof. & Head, Dept. of Mathematics, College of Commerce, Arts & Science,Patna-20. 2 Research Scholar ,Dept. of Mathematics, M. U. Bodh Gaya