DEĞİŞMELİ OLMAYAN HALKALARDA COHEN VE KAPLANSKY TEOREMLERİNİN GENELLEMELERİ

Abstract

In commutative setting, prime ideals are very important tools to determine the structure of a ring. In this thesis, some structure theorems will be discussed which belong to Cohen and Kaplansky. The aim of this thesis is to examine the noncommutative generalizations of Cohen and Kaplansky Theorems, especially considering Reyes’s works in 2010 and 2012. The introductory chapter consists of informations about the importance and the historical improvement of the thesis subject. The second chapter contains basic information needed throughout the thesis. In the third chapter, Cohen and Kaplansky Theorems and their roles in commutative rings are emphasized. Also, S-Noetherian ring structure which was defined by Anderson and Dumitrescu is introduced and some features of this structure are indicated. In the fourth chapter, one-sided generalizations of prime ideals in noncommutative settings are examined and some concepts like completely prime ideals and Oka families are described. Their role in the structure of a noncommutative ring is examined with applications. The fifth chapter is concerned with the noncommutative generalizations of Cohen and Kaplansky Theorems by the Oka families and the point annihilator sets. In the last chapter, noncommutative generalizations of Cohen and Kaplansky Theorems obtained by different approaches are investigated. Among the generalizations discussed by Koh, Chandran and Michler, the noncommutative generalization of S-Noether ring structure is also examined.