The additive subgroups generated by a polynomial

This thesis essentially consists of four chapters. In the first chapter, information concerning the subject of the thesis and works which are related to this subject are shortly given. In the second chapter, firstly basic definitions and theorems related to polynomials and additive subgroups are involved. Later some definitions and theorems which will be used in the thesis have been given. In the third part of the additive subgroup generated by a set of values of a polynomial on two-sided any ideal of a prime rings were examined. The fourth section is the additive subgroup generated by set of values of a polynomial over a right prime ideal of the prime ring were examined.\