Document Type : Original Paper


Department of Mathematics, Faculty of Basic Sciences,Babol University of Technology, Babol, Islamic Republic of Iran


In this paper we introduce a generalization of M-small modules and discuss about the torsion theory cogenerated by this kind of modules in category . We will use the structure of the radical of a module in  and get some suitable results about this class of modules. Also the relation between injective hull in  and this kind of modules will be investigated in this article.   For a module  we show that N is M-Rad if and only if ; where  is the M-injective hull of N.  We will show that for a cohereditary module M,R[M] is closed under extension.  Let  be a module and , the torsion theory cogenerated by  is the reject of  in , defined as .  In this paper we study about the property of this torsion theory.  We show that    if and only if for every nonzero homomorphism  in , .  Another attractive result is  if and only if , for all . For a module  we show that if  for some , then the inclusion  is M-coRad and also if , then for every submodule  of  and M-coRad inclusion , we have .  Finally for a pseudo projective module M we show that every  with  is M-Rad and if moreover , then .