Subgroup, submodule and ideal relationships with the help of augmentation ideals in group rings

The aim of this thesis study is to investigate the correspondences and relations between the subgroups of the group on which the group ring is defined, the sub-modules defined on the group ring and the ideals of the group ring, with the help of augmentation ideals in group rings. For this purpose, the thesis study consists of five chapters. The first part is the introductory part. In this section, the structure of group rings and the historical development of algebraic structures that emerged with the development of group rings are given. The second section is the general information section. In this section, the basic definitions and theorems related to group rings are given. Then, the augmentation ideal of RG was examined in order to see the relationships between the sub-groups of G and the ideals of RG . Also, polynomial ideals in group rings and augmentation ideal as a polynomial ideal are discussed. Finally, basic definitions and theorems about augmentation annihilators are given in order to examine the relations of augmentation ideals with group rings. The third section is the material and method section. This chapter begins by examining normal subgroup congruences. The agreement between the normalized isomorphism and the class sums of the finite G and H groups is shown and definitions and theorems showing the relationship between the lattice of normal subgroups of G and the lattice of normal subgroups of H are given. In the remainder of the chapter, some basic characterizations between the augmentation ideal of a group ring are given and the isomorphism problem for the group and the class of metabelian groups in which the group ring is defined are examined. The fourth section is the research findings and discussion section. In this section, some relationships between R −module classes and RG −module classes are examined. In addition, the similarities and relationships between normal subgroups of G and RG - submodules and between RG −submodules and a normal subgroup of G were examined. The fifth chapter is the conclusion part in which the thesis is summarized.