This thesis is based on the article “Direct Sums of Rickart Modules” written by G.Lee, S.T. Rizvi and C.S. Roman. Already, the concept of Rickart ring was defined by Kaplansky. Suppose is a unitary ring. is called right Rickart, if the right annihilator of every element of , as a right ideal, generates by an idempotent element of . Lee, Rizvi and Roman introduced the notion of Rickart modules motivated by a need to put the notion of right Rickart rings in a general module theoretic setting and by the question: “If is a right Rickart ring and , what kind of Rickart property will the right -module have?” Assume is a right -module. is called Rickart if for every endomorphism of , . It has been shown that every direct summand of a Rickart module is a Rickart module, but examples show that a direct sum of Rickart modules is not always Rickart. In this thesis, we consider this question: “When are the direct sums of Rickart modules, also Rickart?”