Automorphisms of Zero Divisor Graphs of Square Radical Zero Commutative Unital Finite Rings

Lao Hussein Mude Lao Hussein Mude,Owino Maurice Oduor Owino Maurice Oduor,Ojiema Michael Onyango Ojiema Michael Onyango

doi:10.24297/jam.v19i.8834

Lao Hussein Mude Lao Hussein Mude, Owino Maurice Oduor Owino Maurice Oduor + Show 1 more

Open Access

https://doi.org/10.24297/jam.v19i.8834

Copy DOI

Abstract

There has been extensive research on the structure of zero divisors and units of commutative finite rings. However, the classification of such rings via a well-known structure of zero divisors has not been done in general. More specifically, the automorphisms of such classes of rings have not been fully characterized. In this paper, we obtain a more completeillustration of the automorphisms of zero divisor graphs of finite rings in which the product of any zero divisor

Highlights

The classification of completely primary finite rings has received much attention in the recent years.For related studies see [1,2,3,4,5,6]
Most scholars have concentrated in obtaining the structures of zero divisor graphs of commutative finite rings
Little research has been done on the automorphisms of zero divisor graphs of completely primary finite rings,despite the fact that automorphisms have played practical role in understanding complexity of many algebraic structures

Summary

Introduction

The classification of completely primary finite rings has received much attention in the recent years. Most scholars have concentrated in obtaining the structures of zero divisor graphs of commutative finite rings. Little research has been done on the automorphisms of zero divisor graphs of completely primary finite rings ,despite the fact that automorphisms have played practical role in understanding complexity of many algebraic structures. In view of the above, this research investigated the automorphsms of zero divisor graphs of square radical zero commutative unital finite rings. R shall denote commutative finite rings as constructed in the sequel, Z(R) will represent the Jacobson radical of R, the automorphism of the zero divisor graph of the ring R will be denoted by Aut(Γ(R)). The cardinality of the automorphism group shall be denoted by |Aut(Γ(R))|. The following result due to Raghavendran will be useful in the sequel

Preliminary Results

The Construction of Square Radical Zero Finite Rings

Characteristic of R is p

Characteristics of R is p2