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=EndR(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, this is not so for direct sums. We obtain conditions which allow direct sums of (dual) Rickart modules to be (dual) Rickart.
It is shown that an indecomposable endoregular module has precisely a division ring as its endomorphism ring. We introduce the notion of endo-nonsingularity of a module using the module’s endomorphism ring. This notion helps us to obtain new module theoretic versions of the Chatters-Khuri’s result.
Our investigations allow us to work in a general setting of module theory where several known results on (right) Rickart rings and Baer modules are generalized – sometimes more effectively.