On the Zero Divisor and Cayley Graphs of Some Classes of the 2-Radical Index of Nilpotence Finite Local Rings

Abstract

The characterization of finite local rings via the well known structures of their zero divisor graphs and cayley graphs remains an open problem. Some classes of completely primary finite rings which are local, have however been characterized by the compartments of their units and zero divisors where the classification of the unit groups have been done using the Fundamental Theorem of finitely generated Abelian Groups while the zero divisors have been characterized via the zero divisor graphs. This paper characterizes the zero divisor graphs Γ(R) and cayley graphs CAY (R) where R is a finite local ring with 2-radical index of Nilpotence. These two classes of graphs have been completely described and compared using their algebraic properties. Some of the graphs have been drawn for purposes of their comparison. The methods of study involved partitioning the ring under consideration into mutually disjoint subsets of invertible elements and zero divisors and determining their graphs using case by case basis discovery of their structural properties. Some symmetric groups associated with the graphs studied have also been given.