Structures of Selected Classes of Five Radical Zero Completely Primary Finite Rings

Abstract:A Galois ring is a ring in which the subset of zero divisors (including zero) forms a principal ideal. These rings are crucial in the development of completely primary finite rings which are necessary for the classification of finite rings. Construction of finite rings in which; the product of any two zero divisors is zero, the product of any three zero divisors is zero and the product of any four zero divisors is zero have been done by authors such as Corbas, Chikunji, Owino and Ojiema among others. The study of the structures of these rings; the group of units and the set of zero-divisors have also been explored. For the constructed classes of finite rings, the group of units has been classified for each of the characteristic while the set of zero-divisors for some rings have led to the characterization of the zero divisor graphs. In the structure theory of finite rings with identity, classification of finite rings into well-known structures is still inconclusive. Therefore, this work has continued with the classification of finite rings by constructing a class of completely primary finite ring in which the product of any five zero divisors is zero through idealization of Galois ring modules. For the selected classes of five radical zero completely primary finite rings, their elements have been identified and properties studied. The units have been classified via the fundamental theorem of finitely generated abelian groups while the zero divisors have been characterized through the algebraic theory of the zero divisor graphs. This has been possible through isolation of set of units from the set of zero divisors. In the classification of unit groups, it has been noted that some results are in piece when and while in characterization of zero divisor graphs, it has been realized that the zero divisor graphs are incomplete, connected and has a girth of three with a diameter of two. These graphical properties have been realized to be invariant for all the characteristics of the ring. This in turn has elucidated the structures of the zero divisors. This work has provided an application on zero divisor graph and recommended a study on automorphisms of unit groups and zero divisors graphs of five radical zero completely primary finite rings as well as the study of the rings whose subsets of zero divisors are of higher indices of nilpotence.

Created with Mobirise

Web Page Builder