In this thesis , we obtain a partial solution to the following question of Kothe [37] : For which rings R is it true that every left (or both left and right) R-module is a direct sum of cyclic modules? Let R be a ring in which all idempotents are central . We prove that , if R is a left Kothe ring (i.e. , every left R-module is a direct sum of cyclic modules) , then R is an Artinian principal right ideal ring . Consequently , R is a Kothe ring (i.e. , each left , and each right , R-module is a direct sum of cyclic modules) if and only if R is an Artinian principal ideal ring . This is a generalization of the Kothe-Cohen-Kaplansky theorem [11] and [37] . Also , an interesting natural question of this sort is " whether the same is true if one only assumes that every ideal is a direct sum of cyclic modules? " In this thesis we answer this question in the case R is a finite direct product of commutative Noetherian local rings . The structure of such rings is completely described . In particular , this yields characterizations of all commutative Artinian rings with this property . Also , in this thesis we study commutative rings R whose prime ideals are direct sums of cyclic modules . In the case R is a finite direct product of commutative local rings , the structure of such rings is completely described . Also , we establish a theorem which state that , to check whether every prime ideal in a Noetherian local ring (R , M ) is a direct sum of (at most n) principal ideals , it suffices to test only the maximal ideal M . Finally , we investigate the structure of cyclically pure (or C-pure) projective modules .In particular , it is shown that a ring R is left Noetherian if and only if every C-pure projective left R-module is pure projective . Also , over a left hereditary Noetherian ring R, a left R-module M is C-pure projective if and only if M = N+ P, where N is a direct sum of cyclic modules and P is a projective left R-module . The relationship C-pure projective modules with pure projective modules and RD-projective modules are also studied . It is shown that if R is a local duo-ring , then the C-pure projective left R-modules and the pure projective left R-modules coincide if and only if R is a principal ideal ring . Also , if R is a left perfect duo-ring , then the C-pure projective left R-modules and the pure projective left R-modules coincide if and only if R is a left Kothe ring . Also , we characterize commutative rings R for which the C-pure projective modules and the RD-projective modules coincide . Moreover , we show that if R is a left p.p-ring and every C-pure projective left R-module is RD-projective , then R is a left Noetherian left hereditary ring . The converse is also true when R is a commutative ring , but it is not true when R is a Non-commutative ring.