Publication: Yarı halkaların asal radikalleri
Abstract
YARI HALKALARIN ASAL RADİKALLERİ Tezin temel amacı halkalarda tanımlanan radikal sınıf ve asal radikal kavram ve özelliklerini değişmeli olmayan yarı halkalar üzerinde tanımlamak ve özelliklerini incelemektir. Bunun için ilk olarak yarı halka kavramı ve temel özellikleri tanıtılmıştır. Yarı halkaların idealleri ve homomorfizmaları ile halka idealleri ve homomorfizmaları arasında ilişkiler kurulmuştur. Bölüm halkalarının cebirsel yapısı gibi bölüm yarı halkaları inşa edilmiştir. Daha sonra halkalardaki radikal teorinin temel yapıları yarı halkalara genişletilmiştir. Bu genişlemeler yardımıyla yarı halkaların yarı basit sınıfları, radikal sınıfların kesişim ve birleşimi incelenmiştir. Çalışmanın son kısımlarında ise halkadaki Lower Bear Radikali, yarı halkalarda tanımlanmıştır. Daha sonra halkalarda radikal teoriye bağlı olarak tanımlanan asal radikal kavramının yarı halkaların Lower Bear Radikaliyle çakıştığı sonucu elde edilmiştir.
THE PRIME RADICALS OF THE SEMIRINGS The main aim of the study is to describe the concept of the radical classes and the prime radical over the intransitive semirings which were previously defined over the rings and to investigate their features. Firstly, the semiring concept and its basic properties were introduced in the study. Relationships were established between the ideals of the semirings, their homomorphisms; and the ideals of the rings, their homomorphisms. Quotient semirings as the quotient rings’ algebraic structure were constructed. Afterwards the main structure of the radical theory on the rings was extended to the semirings. Semisimple classes of semirings, union and intersection of radical classes were examined with the assistance of these expansions. At the end of the study, the Lower Bear Radical for rings was identified in semirings. Then as a result, it was concluded that there was a coincidence between the concept of prime radical defined depending on the radical theory for rings and the Lower Bear Radicals for semirings.
THE PRIME RADICALS OF THE SEMIRINGS The main aim of the study is to describe the concept of the radical classes and the prime radical over the intransitive semirings which were previously defined over the rings and to investigate their features. Firstly, the semiring concept and its basic properties were introduced in the study. Relationships were established between the ideals of the semirings, their homomorphisms; and the ideals of the rings, their homomorphisms. Quotient semirings as the quotient rings’ algebraic structure were constructed. Afterwards the main structure of the radical theory on the rings was extended to the semirings. Semisimple classes of semirings, union and intersection of radical classes were examined with the assistance of these expansions. At the end of the study, the Lower Bear Radical for rings was identified in semirings. Then as a result, it was concluded that there was a coincidence between the concept of prime radical defined depending on the radical theory for rings and the Lower Bear Radicals for semirings.