Theory of Rickart Modules

This dissertation is devoted to investigations in the theory of Rickart modules.We introduce various notions related to the Rickart property in a general module theoretic setting.Endomorphism ring of a module plays an important role in our study.Topics of our study include: Rickart modules, dual Rickart modules, endoregular modules and endo-Rickart modules.These notions constitute the main chapters of the dissertation.A module M is called Rickart if the right annihilator in M of any single element of S = End R (M ) is generated by an idempotent of S. This extends the notion of a Baer module as well as that of a right Rickart ring.M is called a dual Rickart module if the image in M of any single element of S is generated by an idempotent of S. M is called an endoregular module if its endomorphism ring is von Neumann regular.M is called an endo-Rickart module if the left annihilator in S of any single element of M is generated by an idempotent in S.We provide several characterizations and investigate properties of each of these concepts.We also study the connections of such modules with their endomorphism rings.It is shown that a (dual) Rickart module whose endomorphism ring has no infinite set of nonzero orthogonal idempotents is a (dual) Baer module.We obtain characterizations of well-known classes of rings R, in terms of Rickart R-modules.While direct summands of (dual) Rickart modules are shown to inherit the property, I thank my advisor, Professor S. Tariq Rizvi, and co-advisor, Cosmin Roman, both of whom have been very helpful and have shared a lot of time with me during my work.Without them, I can not be one of mathematicians