www.ccsenet.org/jmr Journal of Mathematics Research Vol. 3, No. 2; May 2011 On Fully-M-Cyclic Modules Samruam Baupradist (Corresponding author) Department of Mathematics, Faculty of Science Chulalongkorn University, Bangkok 10330, Thailand Tel: 66-83-137-1119 E-mail: samruam.b@chula.ac.th Suphawat Asawasamrit Department of Mathematics, Faculty of Applied Science King Mongkut ’s University of Technology North Bangkok Bangkok 10800, Thailand Tel: 66-81-346-6321 E-mail: suphawata@kmutnb.ac.th Received: November 26, 2010 Accepted: December 7, 2010 doi:10.5539/jmr.v3n2p23 Abstract The aim of this work was to generalize generator, M-generated modules in order to apply them to a wider class of rings and modules. We started by establishing a new concept which is called a fully- M-cyclic module. We defined this notation by using Hom R ( M, ∗) operators which are helpful to contract the new construction and describe their properties. Finally, we could see the structure of fully- M-cyclic module and quasi-fully-cyclic module by the structure of M. Keywords: Fully- M-cyclic modules, Quasi-fully-cyclic modules, Generator modules, Self-generator modules 1. Introduction Throughout this paper, R is an associative ring with identity and M R is the category of unitary right R-modules. Let M be a right R-module and S = End R ( M), its endomorphism ring. A right R-module N is called M-generated if there exists an epimorphism M (I ) −→ N for some index set I . If I is finite, then N is called finitely M-generated. In particular, N is called M-cyclic if it is isomorphic to M/L for some submodule L of M. Following Wisbauer [1991], σ[ M] denotes the full subcategory of Mod-R, whose objects are the submodules of M-generated modules. A module M is called a self-generator if it generates all of its submodules. M is called a subgenerator if it is a generator of σ[ M]. 2. On Fully- M-cyclic module In this part, a module M be given as a right R-module. Definition 2.1. Let N ∈ M R . N is called a fully- M-cyclic module if every submodule A of N is of the form s( M) for some s in Hom R ( M, N). Remark 2.2. Dealing directly from definition, the following statements are routine: (1) Submodule of a fully- M-cyclic module is a fully- M-cyclic module. (2) If M is simple module and N is fully- M-cyclic module, then any nonzero submodule of N is simple submodule. Definition 2.3. The module M ∈ M R is called a quasi-fully-cyclic module if it is a fully- M-cyclic module. Obviously, every semi-simple module is a quasi-fully-cyclic module. Lemma 2.4. Let N be a fully-M-cyclic module. If M is a noetherian module then S oc( M)  S oc(N). Proof. Since N is a fully- M-cyclic module, a simple submodule B of N is of the form s( M) for some s ∈ Hom R ( M, N). By the simply property of B, there is b ∈ B such that B = bR. Suppose that s(a) = b for some a ∈ M. In noetherian module aR, there exists a simple submodule A containing a. It is easily to see that A  B. Conversely, if A is a simple submodule of M then s(A) = B is a simple submodule of N and then A  B for all s ∈ Hom R ( M, N). This shows that S oc( M)  S oc(N). ✷ Lemma 2.5. If N is a fully-M-cyclic module then N has no nonzero small submodule. Proof. In a contrary, we suppose that there is a nonzero submodule A which is small in N. Let B be a submodule of N Published by Canadian Center of Science and Education 23