Let R be a commutative ring with identity, and let M be a unitary (left) R- modul e. The ideal annRM  = {r E R;rm  = 0 V  mE M} plays a central

role  in  our  work.  In  fact,  we  shall  be  concemed   with  the  case  where annR1i1 = annR(x) for   some   x EM such  modules  will  be  called bounded  modules.[t  htrns out that there are many classes of modules properly contained in the class of bounded modules such as cyclic modules, torsion -G·ee modulcs,faithful  multiplication  modules,  prime modules and cyclic modules over  their endomorphism  rings. Also,  using  boundedness of  modules,  we showed that :

-  The  classes  of  injective modules modulo  annihi lator  and  quasi-injective

modules ru·e equivalent.

-  The  classes  of   faithful  modules  and  compactly   faithful  modules  are equi valent.