Structure of rings with conditions on certain subsets -

Abstract

Rings are of different structures, altering some conditions on some subsets of a ring might cause a change in its structure. In this thesis, we study some of these alterations and their effects on the ring and its subsets. In the first chapter, we introduce some basic definitions, theorems, and lemmas that are crucial for the succeeding chapters. In the Second chapter, we study the structure of rings with prime centers and how varying some conditions on these rings affects its commutativity. We then conclude this chapter by giving an example showing that a ring with a prime center is not necessarily commutative. In the third chapter, we show that rings multiplicatively generated by idempotents and nilpotents might undergo a change in its structure and the structure of some of its subsets when it’s given some conditions. Furthermore, we show that a ring R which is either finite or has an identity is necessarily Boolean when given some property; we then give an example which shows that the finiteness of R and the existence of its identity along with this property are essential to prove that R is Boolean. In the final chapter, we show how putting some conditions on subsets that are multiplicatively generated by idempotents and nilpotents of a ring might alter the structure of the ring and its subsets.