Zero Divisor Graphs Of Classes Of Completely Primary Finite Rings Of Maximal Prime Power Characteristic

Abstract

It is well known that in a nite ring with identity, every element is either a zero

divisor or a unit. The classication of nite rings is not fully settled. Dierent

studies have generated interesting results on certain classes of nite rings. It is

worthwhile to note that completely primary nite rings have proved to be useful

towards the classication of nite rings. This is due to the fact that a nite ring

has a unique maximal ideal if and only if it is a full matrix ring over a completely

primary nite ring. Moreover, any commutative ring is a direct sum of completely

primary nite rings. A deeper understanding of the elements in a nite ring enables

us to fully understand the ring. In this study, we investigate and characterize the

zero divisor graphs of classes of commutative completely primary nite rings of

maximal prime power characteristic. For each class of rings, zero divisor graphs are

drawn and trends in their geometric properties established through graph theoretic

approach. In higher order cases, properties of zero divisors of commutative rings

are employed in interpreting and determining the invariant geometrical structures of

the graphs. This study has established that the diameter of the zero divisor graphs

of the rings studied lie between 0 and 2 while their girth is either 3 or 1: None

of the rings has a zero divisor graph that is n-gon, where n is an integer greater

than 3. Fundamentally, this study has revealed that rings whose zero divisor graphs

are isomorphic are not necessarily isomorphic. The ndings of this study extend

further the knowledge about the structure theory of nite rings and in particular,

the classication of the zero divisors of commutative completely primary nite rings.