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.
