Department of Mathematics, Vali-e-Asr University of Rafsanjan, P.O. Box 7718897111, Rafsanjan, Iran.
Let $R$ be a commutative ring with identity and $M$ be a finitely generated unital $R$-module. In this paper, first we give necessary and sufficient conditions that a finitely generated module to be a multiplication module. Moreover, we investigate some conditions which imply that the module $M$ is the direct sum of some cyclic modules and free modules. Then some properties of Fitting ideals of modules which are the direct sum of finitely generated module and finitely generated multiplication module are shown. Finally, we study some properties of modules that are the direct sum of multiplication modules in terms of Fitting ideals.
 I. Akharraz, M. E. Charkani, Induced modules by endomorphism of nitely generated modules, Int. J. Algebra 3(12) (2009), 589-597.  A. Barnard, Multiplication modules, J. Algebra 71 (1981), 174-178.  W. C. Brown, Matrices Over Commutative Rings", Pure Appl. Math. 169, Marcel Dekker Inc., New York, 1993.  D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry", Springer-Verlag, New York, 1995.  Z. A. El-Bast, P.F. Smith, Multiplication modules, Comm. Algebra 16 (1988), 755-779.  N. S. Gopalakrishnan, Commutative Algebra", Oxonian Press, New Delhi, 1984.  J. Ohm, On the rst nonzero Fitting ideal of a module, J. Algebra 320 (2008), 417-425.  J. Rotman, An Introduction to Homological Algebra", Springer, 2008.  Y. Tiras, M. Alkan, Prime modules and submodules, Comm. Algebra, 31 (2003), 5253-5261.
 P. Vamos, Finitely generated Artinian and distributive modules are cyclic, Bull. London Math. Soc. 10 (1978), 287-288.