Nilpotent, idempotent and units in group rings

This thesis consists of five chapters. In the first chapter, some basic definitions and theorems which are necessary for next chapters, are introduced. In the second chapter, some related studies in the literature are given widely. In the third chapter, materials and methods which are used in the thesis are given. In the fourth chapter, some results, definitions and theorems which are originally found are introduced by findings as three subsections. In the first of these subsections, some necessary and sufficient conditions for all the normalized units in commutative group rings of some direct product groups to be G×H-nilpotent units are given. In the second subsection, idempotent units are defined in the commutative group rings of some direct product groups and some necessary and sufficient conditions for every normalized unit in commutative group rings of some direct product groups to be an idempotent unit are obtained. In the third subsection, unit group of integral group ring of the direct product group S_3×C_3 is characterized in terms of the torsion-free normal complement of S_3, where S_3 is a symmetric group of order 6 and C_3 is a cyclic group of order 3. Fifth and the last chapter comprises of results, discussion on the thesis and suggestions on some open problems.