A study on modules whose endomorphism rings are regular

In the first part of our thesis, a brief information about the studies on regular rings is given. In the second chapter, the general properties of regular and unit regular rings, and some basic concepts that we will use throughout the thesis are given. In the third part of the thesis, modules with regular endomorphism rings, called endoregular, are examined in details. These modules are compared with Zelmanowitz regular modules. For some special ring classes, characterizations are handled in terms of endoregular modules. It is examined with examples that the feature of being endoregular need not carry over to the direct sum, and it is investigated when the direct sum of endoregular modules is endoregular. In addition, the endomorphism rings which satisfy some other regularity conditions are also studied. In the fourth chapter, modules whose endomorphism ring is unit regular, namely unit endoregular modules are discussed. The characterizations given for these modules are examined. In parallel with the endoregular modules, it is investigated when the direct sum of the unit endoregular modules is unit endoregular.