en عمومى General پژوهشي Research paper <p dir="ltr">In this paper, by considering the notion of extended BCK-module, we define the concepts of free extended BCK-module, free object in category of extended BCK-modules and we state and prove some related results. Specially, we define the notion of idempotent extended BCK-module and we get some important results in free extended BCK-modules. In particular, in category of idempotent extended BCK-modules, we give a method to make a free object on a nonempty set and in BCK-algebra of order 2, we give a method to make a basis for unitary extended BCK-modules. Finally, we define the notions of projective and productive modules and we investigate the relation between free modules and projective modules. In special case, we state the relation between free modules and productive modules.</p> BCK-algebra, Extended BCK-module, Free extended BCK-module. 29 43 http://ijmsi.ir/browse.php?a_code=A-10-523-1&slc_lang=en&sid=1 R. A. Borzooei borzooei@sbu.ac.ir 10031947532846006221 10031947532846006221 Yes Department of Mathematics, Shahid Beheshti University S. Saidi Goraghani SiminSaidi@yahoo.com 10031947532846006222 10031947532846006222 No Department of Mathematics, Islamic Azad University of Central Tehran