East Asian Mathematical Journal Vol. 27 (2011), No. 1, pp. 75–82 ON FUZZY PRIME SUBMODULES OF FUZZY MULTIPLICATION MODULES Dong Soo Lee and Chul Hwan Park Abstract. In this paper, we will introduce the concept of fuzzy mulit- plication module. We will define a new operation called a product on the family of all fuzzy submodules of a fuzzy mulitplication module. We will define a fuzzy subset of the idealization ring R+M and find some relations with the product of fuzzy submodules and product of fuzzy ideals of the idealization ring R + M. Some properties of weakly fuzzy prime submod- uels and fuzzy prime submodules which are defined by T.K.Mukherjee, M.K.Sen and D.Roy will be introduced. We will investigate some prop- erties of fuzzy prime submodules of a fuzzy multiplication module. 1. Introduction In this paper we will investigate some properties of fuzzy multiplication modules. In section 2, we will review some properties of fuzzy ideals of a ring and fuzzy submodules of a module. Since A.Barnard investigated some properties of multiplication modules and ideals via the paper which was published in J of Algebra, several authors such as P. F. Smith, Z. El-Bast, R. Ameri and S. Atani found various properties of multiplication modules through some papers. In section 3, We define a fuzzy multiplication module by using the concept of fuzzy submodules of an R-module M . An R-module M is called a fuzzy multiplication module iff for each fuzzy submodule U of M : there exists a fuzzy ideal A of R such that U =A1 M . R. Ameri defined a product of two submodules on the family of all submod- ules of a multiplication module. In this paper we will define a product of fuzzy submodules on the family of all fuzzy submodules of M (denoted by F (M )) as UV =(AB)1 M where U and V are fuzzy submodules of M and A and B are fuzzy ideals of a ring R such that U = A1 M and V = B1 M . We will investigate some properties of this product and find some relations with respect to the ring which is the idealization ring given by R + M . Received September 2, 2010; Accepted January 3, 2011. 2000 Mathematics Subject Classification. 03E72,03F55,13C99. Key words and phrases. Fuzzy multiplication module, the idealization of module, fuzzy prime submodule. The first author was supported by University of Ulsan Research Fund of 2010. c 2011 The Youngnam Mathematical Society 75