Eversible rings and zero-divisors

Abstract

This thesis centers on the study of eversible rings, a class of rings in which every onesided zero-divisor is necessarily a two-sided zero-divisor. The concept of eversibility generalizes the idea of reversibility and offers a new perspective on the structure of noncommutative rings. To motivate this study, Chapter 1 provides a historical and conceptual overview of zero-divisors and their significance in both commutative and noncommutative settings. Chapter 2 introduces some standard rings such as directly finite rings, von Neumann regular rings, trivial extensions and skew polynomial rings, which lay the groundwork for the main investigation. In Chapter 3, the focus shifts to the study of zero-divisors and eversibility in specific ring contexts, including formal triangular matrix rings, upper triangular matrix rings, polynomial rings and formal power series rings. The final section of Chapter 3 critically addresses several incorrect results from previous studies on eversibility. Through carefully constructed counterexamples, the thesis disproves these erroneous claims, further refining our understanding of the conditions under which eversibility holds in these contexts.