This thesis is based on an article by Gangyong Lee, S. Tariq Rizvi and Cosmin S. Roman in 2010 . We assume that R is a ring (not necessarily commutative) with unity and M is an unital right R -module. The concept of right Rickart rings (Maeda,1960) or right p.p. rings (Hattori,1960) has been extensivly studied in the literature. A ring R is called right Rickart if the right annihilator of any single element of R is generated by an idempotent as a right ideal. A left Rickart ring is defined similarly. notion of Rickart rings is not left-right symmetric. Examples of right Rickart rings include domain, von Neumann regular rings and right (semi) hereditary rings. In particular, the endomorphism ring of an arbitrary direct sum of copies of a right hereditary ring is a right Rickart ring. Let S= (M) be the ring of R -endomorphisms of M . A right R -module M is called a Rickart module if the right annihilator in M of any single element of S is generated by an idempotent of S , equivalently, ( )= Ker( ) M for every . It is easy to see that for M= , the notion of a Rickart module coincides with that of a right Rickart ring. It is shown that every direct summand of a Rickart module inherits the property. Every Rickart module is K -nonsingular and has the Summand Intersection Property (SIP). The type="#_x0000_t75" (M) is a right Rickart ring and M is k -local-retractable. In 1967, Small proved that a right Rickart ring with no infinite set of nonzero orthognal idempotents in its endomorphism ring is precisely a Baer module. Analogous to the well known result of small, shown that a Rickart module with no infinite set of nonzero orthogonal idempotents in its endomorphism ring is precisely a Baer module. An arbitrary direct sum M of cyclic submodules over a commutative Dedekind domain is a Rickart module if and only if M is either semisimple or torsion-free, if and only if, S is a right Rickart ring. Also a countable direct sum of finitely generated modules over a principal ideal domain is Rickart if and only if it is either semisimple or torsion-free.
