Small subhypermodules and their applications Behnam Talaee Department of Mathematics, Faculty of Basic Sciences, Babol University of Technology, Babol, Iran behnamtalaee@nit.ac.ir Abstract Let R be a hyperring (in the sense of [5]) and M be a hypermodule on R. In this article we introduce class of small subhypermodules of M . First we get some properties of subhypermodules and then the class of small subhypermodules and small homomorphism in the category of hypermodules are investigated. For example we show that if M is a hypermodule and N is a direct summand of M , then a small subhypermodule K of M which is contained in N , is small in N . Also we get some important applications of small subhypermodules in category of hypermodules (for example in exact sequences etc.). 2010 AMS Classification: 16D80, 20N20 Keywords: hyperring, hypermodule, superfluous subhypermodule, superfluous epimorphism 1 Introduction The categories of hypergroups, hypermodules and hyperrings have many important roles in hyperstructures. Some authors got many exiting results about these theories. Reader can see references [1], [3], [4], [5] to get some basic information about the categories of hypergroups, hyperrings and hypermodules. Also reference [8] can be suitable to get some information about rings and modules theory. We recall some definitions and theorems from above references which we need them to develop our paper. A hyperstructure is a nonvoid set H together with a function . : H × H −→ P ∗ (H ), where . is called a hyperoperation and P ∗ (H ) is the set of all nonempty subsets of H . For A, B ⊆ H and x ∈ H we define A.B =  a∈A,b∈B a.b, x.B = {x}.B, A.x = A.{x}. 1 ROMANIAN JOURNAL OF MATHEMATICS AND COMPUTER SCIENCE, 2013, VOLUME 3, ISSUE 1, p.5-14