## ABSTRACT

This research work deals with commutativity theorems for rings and near-rings with their various generalizations in the setting of some special classes of rings as well as derivations in rings and near-rings. We present their fundamentals and also prove some commutativity results for semi prime rings and ring with unity.We then extended the results to s-unital rings. Some results on commutativity of rings and near-rings are generalized and justified by examples. Posner‟s first theorem isestablished for 𝜎−prime rings. Jordan left and right 𝜎−derivations as well as generalized Jordan left and right 𝜎−derivations on 𝜎−rings are investigated. Further more,distributive near-ring, pseudo-abelian near-ring and distributively generated near-rings are introduced andthese help in developing certain class of derivations on near-rings. Finally, permuting 4−(𝜎,𝜏) derivations on prime near-rings are generalized to permuting 𝑛−(𝜎,𝜏) derivations.

## TABLE OF CONTENTS

Cover Page…………………………………………………………………………………………………………….. i

Fly Leaf ……………………………………………………………………………………………………………….. ii

Title Page …………………………………………………………………………………………………………….. iii

Declaration ………………………………………………………………………………………………………….. iv

Certification ………………………………………………………………………………………………………….. v

Dedication……………………………………………………………………………………………………………. vi

Acknowledgements ………………………………………………………………………………………………. vii

Abstract ………………………………………………………………………………………………………………. ix

Table of Contents …………………………………………………………………………………………………… x

CHAPTER ONE: …………………………………………………………………………………………………. 1

GENERAL INTRODUCTION ………………………………………………………………………………. 1

1.1 Introduction ………………………………………………………………………………………………… 1

1.2 Statement of the Problem ………………………………………………………………………………. 5

1.3 Justification of the Study……………………………………………………………………………….. 5

1.4 Aim and Objectives of the Study …………………………………………………………………….. 6

1.5 Methodology of the Study ……………………………………………………………………………… 7

1.6 Definition of Terms ……………………………………………………………………………………… 7

1.7 Organization of the Thesis …………………………………………………………………………… 11

CHAPTER TWO: ………………………………………………………………………………………………. 12

LITERATURE REVIEW ……………………………………………………………………………………. 12

CHAPTER THREE: …………………………………………………………………………………………… 20

FUNDAMENTALS OF COMMUTATIVITY THEOREMS FOR RINGS ……………….. 20

3.1 Ring-theoretic concepts ………………………………………………………………………………. 20

3.2 Results on Semiprime Rings ………………………………………………………………………… 28

3.3 Some Results on Ring with Unity …………………………………………………………………. 31

3.4 Some Results on s-unital Rings …………………………………………………………………….. 42

3.5 Results for Some Special Classes of Rings ……………………………………………………… 45

CHAPTER FOUR:……………………………………………………………………………………………… 56

DERIVATIONS ON SOME SPECIAL CLASS OF RINGS ……………………………… 56

4.1 Definitions of Some Rings and Derivations …………………………………………………….. 56

4.2 Posner’s First Theorem forprime Rings ……………………………………………………. 68

xi

CHAPTER FIVE: ………………………………………………………………………………………………. 81

RESULTS ON SOME SPECIAL CLASSES OF NEAR-RINGS ……………………………… 81

5.1 Near-ring Theoretic Concepts ………………………………………………………………………. 81

5.2 Some Results on Near-rings …………………………………………………………………………. 87

5.3 Derivation in Near-rings ……………………………………………………………………………… 91

5.4 Permuting 4-Derivation on Near-rings …………………………………………………………… 95

5.5 Some Results on Permuting𝒏− 𝝈,𝝉 Derivation on Near-rings ………………………. 105

CHAPTER SIX: ……………………………………………………………………………………………….. 126

SUMMARY, CONCLUSION AND RECOMMENDATIONS ……………………………….. 126

6.1 Summary ………………………………………………………………………………………………… 126

6.2 Conclusion ……………………………………………………………………………………………… 127

6.3 Recommendations ……………………………………………………………………………………. 128

REFERENCES …………………………………………………………………………………………………. 129

## CHAPTER ONE

GENERAL INTRODUCTION

1.1 Introduction

An algebraic system is a non-empty set together with one or more binary operations

defined on the set. This system subsequently considered shall be divided in to

structures according to the axioms, which they satisfy. The division leads to the

branch of mathematics known as abstract algebra (viz: semi group,group, ring, nearring

and so on).Sets and binary operations (functions) are the fundamental and main

ingredients of algebraic systems. Examplesof algebraic systems are setof all natural

numbers (or counting numbers) ℕ, set of allintegers ℤ,set of all rational numbersℚ, set

of all real numbers ℝ and set of all complex numbers ℂ. Each of these sets is an

important algebraic system under addition and multiplication .

A ring is a non empty set with two binary operations, namely addition (+) and

multiplication , satisfying a collection of axioms. These axioms require addition to

satisfy the axioms for an abelian group while multiplication is associative. The two

operations are connected by the (left and right) distributive laws. Some examples of

rings are set of integers, rational numbers, real numbers, complex numbers, 22

real (complex) matrices and so on.

An additively written group 𝑁𝑅 ,+ (not necessary abelian) equipped with a binary

operation (∙) ∶ 𝑁𝑅 × 𝑁𝑅 → 𝑁𝑅 defined by . 𝑥, 𝑦 → 𝑥𝑦 suchthat xy z x yz and

2

𝑥 𝑦 + 𝑧 = 𝑥𝑦 + 𝑥𝑧 ∀ 𝑥, 𝑦, 𝑧 ∈ 𝑁𝑅 is called a left near-ring (respectively, a right nearring

if 𝑦 + 𝑧 𝑥 = 𝑦𝑥 + 𝑧𝑥 ∀ 𝑥, 𝑦, 𝑧 ∈ 𝑁𝑅). A near-ring 𝑁𝑅 is said to be zero

symmetric if 0x 0 for all𝑥 ∈ 𝑁𝑅.

For example, let𝑉 be a linear space with a basis 1 2 e ,e ,. . . , n e over a field 𝐹 of

characteristic different from two. Define a mapping :V V V by vw 0

forall𝑣, 𝑤 ∈ 𝑉 with v ± 1 e and 1 1 e w w, e w w. With respect to this

multiplication𝑉is a left zero-symmetric near-ring. In view of the above multiplication,

1 2 3 2 3 e e e e e . On the other hand, neither 2 3 1 e e e nor 2 3 1 e e e

since𝑒1, 𝑒2, … 𝑒𝑛 is linearly independent, and hence 2 3 1 e e e 0. Since right

distributive law fails 𝑉is not a ring.

The commencement of the twentieth century marks the beginning of the

investigations of classes of rings which turn out to be commutative under

certain types of hypotheses. Although, the famous Weddurburn theorem (1905)

on commutativity of finite division rings was proved in the early twentieth

century, yet it was during the past four decades that striking results with

conditions imposed on a ring, render the ring commutative or almost

commutative. An early example is the Weddurburn theorem on finite division

rings which plays an important role in commutativity theorems.

Much significant contributions were made in (Herstein, 1968) by generalizing

some classical commutativity theorems and obtaining various conditions

which rendered rings to be commutative or nearly commutative. For

instance,let 1 2 , ,…, n Z x x x denote the ring of polynomials over the set of

integers ℤ, in non-commut ing variables 1 2 , , …, . n x x x Let 𝑅 be a ring (not

3

necessarily unital, not necessarily commutative). We say 1 2 , ,…, n f Z x x x is a

polynomial identity (PI) on 𝑅 if 1 2 , ,…, 0 . n i f a a a a R 𝑅 is called PI- ring if

it satisfies a non-trivial polynomial identity. Commutative rings, for example,

all sat isfy the polynomial ident ity xy yx 0,x, yR.

Commutativity theorems provide conditions on a ring 𝑅 which imply that 𝑅 is

commutative.For example, let 𝑅 be a ring for which aR an integer

n(a) 1 with ( ) . n a a a Then 𝑅 is commutative,according to Jacobson‟s theorem

(Jacobson, 1945).Many of the commutativity theorems discussed in (Herstein, 1968)

assert that if 𝑅 satisfies a certain polynomial identities then 𝑅 is commutative.

There is no clear-cut way to allow cancellation among the elements of rings,

which is always permissible in case of groups. This restricted the progress of

rings till the general structure theory (called structure of rings) was developedby

Jacobson (1964). Commutativity theorems are part of the study of polynomial

identities in noncommutative rings. These theorems assert that under certain

conditions, the ring at hand must be commutative. There are several results

dealing with conditions under which a ring is commutative. Generally such

conditions are placed on the ring itself or its subsets, or commutators.

Herstein (1953) proved that commutativity follows if one assumes for every 𝑥

there existsa polynomial 𝑝 with integral coefficients such that 2 x p x x is

central. Martindale (1958) went still further by weakening the assumption to the

existence of an element 𝑥 such that 2 x a x is central.

4

If every part of a ring 𝑅 satisfies:all commutators arecentral, all nilpotent

elements are central, andall one-sided ideals are two sided, then 𝑅 is

commutative. It is then natural to look at the extension of commutativity of rings

through the following properties:

a. To prove commutativity of 𝑅, a ring with unity 1, if 𝑅 is n-torsion free and

g X ,hX Z X such that

, ,

k

p n q t x x y y f y x y g y h y ∀𝑥 ∈ 𝑅,where p 0, q 0, t 0,n >1, k >1 are

fixed integers.

b. For each 𝑥, 𝑦 ∈ 𝑅, there exist polynomials 𝑓 𝑡 , 𝑔(𝑡) ∈ 𝑍 𝑡 and integers

𝑟 = 𝑟 𝑥, 𝑦 > 1, 𝑠 = 𝑠 𝑥, 𝑦 > 1, 𝑝 = 𝑝 𝑥, 𝑦 ≥ 0, 𝑞 = 𝑞 𝑥, 𝑦 ≥ 0, 𝑚 ≥ 1,

𝑡 ≥ 2with 𝑟, 𝑠 = 1 such that , p r x x y ,

m

t g y x y f y h y and

~ ~

, ,

m

q s t x x y g y x y f y h y .

Remark1.1

In problem a;

(i). one can examineevidently that there are dual properties found by

replacing , p n q x x y y by , p n q y x y x in definition (a)., and also by

replacing , p r x x y and , q s x x y by , r p x y x and , s q x y x in (b).

(ii). rings with unity and s-unital rings are considered to satisfy properties

(a) and (b) above for the development of commutativity theorems.

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1.2 Statement of the Problem

The introduction of rings theory is a boostto modern algebra.However, the inadmissibility of cancellation law impeded the early development of ring theory until the structure theory of rings was propoed by Jacobson (1964).Many researchers have contributed to the concepts of commutativity theorems of rings and near rings in recent time. Prominent among the contributors are Jing and Lu (2003), Oukhtite (2011), Pinter-Lucke (2007), Vukman (2007), Dhara and Shrama (2009), Khan et al. (2013), Huang (2011), Ashraf and Siddeeque (2013) and so on. Despite their giant strides, the notion of permuting 4− 𝜎,𝜏 derivation as well as its generalization to permuting 𝑛− 𝜎,𝜏 derivations in commutative ring and near ring are still unexplored.

Motivated by these, we introducethe notion of permuting 4− 𝜎,𝜏 derivation as well as permuting 𝑛− 𝜎,𝜏 derivation in near-rings, where n is a positive integer.We study some properties of the derivations and show that additive commutativity of a near ring 𝑁𝑅 satisfies certain identities involving permuting 𝑛− 𝜎,𝜏 derivations of a prime near ring. We alsogive illustrations to justify the notions in various theorems, and investigate the polynomial identities with constraints such as commutators, torsion free conditions on rings and near-rings to establish commutativity theorems.

1.3 Justification of the Study

Commutativity theorems for rings and near-rings with their applications have been studied by lots of Mathematicians. The essence of this research is to encourage the pursuit of research on applications of ring theory in diverse areas; such as toconcentrate on the interdisciplinary efforts involved in the pursuit of information

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technology and coding theory. Boolean rings are well-known for their applications in various branches of mathematics as well as in computer science. All such rings are in fact commutative rings. Near-ring theory is applicable in computer science, digital computing, data communications, and so on. Hence the study of commutativity of rings is an important problem in non commutative rings.

1.4 Aim and Objectives of the Study

The aim of this research work is to investigate some results on commutativity of semi prime rings, rings with unity, s-unital rings and permuting 4− 𝜎,𝜏 derivation as well as permuting 𝑛− 𝜎,𝜏 derivations on prime near-rings. In order to achieve the above aim,the objectives considered are to: (i). extend the related results for one sided s-unital rings and n-torsion free rings,

(ii). establish the resultsof Jordan right 𝜎−derivation and generalized Jordan right

𝜎−derivation on 𝜎− rings,

(iii). introduce the notion of permuting4− 𝜎,𝜏 derivation as well as permuting

𝑛− 𝜎,𝜏 derivation in near-rings,

(iv). show that additive commutativity of a near ring 𝑁𝑅 satisfies certain identities

involving permuting 𝑛− 𝜎,𝜏 derivations of a prime near ring,

(v). give examples to justify the notions of permuting4− 𝜎,𝜏 derivation and

permuting 𝑛− 𝜎,𝜏 derivations,

(vi). extend Posner‟s first theorem to 𝜎−prime rings of characteristic different fromtwo, (vii). examine polynomial identities with constraints such as commutators, torsion free conditions on rings and near-rings thereby establishing commutativity

theorems.

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1.5 Methodology of the Study

In order to achieve the objectives outlined above, we conducted an up-to date critical

review of literature dealing with various fundamental aspects of commutativity of

rings and near-ring, and define some terms on derivations in near-rings which helpin

formulatingour results.

1.6 Definition of Terms

a. A map 𝑑: 𝑅 → 𝑅 is called a derivation of ring 𝑅 if it satisfies the following

properties:(i). 𝑑 (𝑥 + 𝑦) = 𝑑(𝑥) + 𝑑(𝑦); and

(ii). 𝑑(𝑥 𝑦) = 𝑦𝑑(𝑥) + 𝑥 𝑑(𝑦), for all 𝑥, 𝑦 ∈ 𝑅.

The example of a non trivial derivation is the usual differentiation on the ring 𝐹[𝑥] of

polynomials defined over a field 𝐹.

b. For a fixed a in 𝑅, define 𝑑a: 𝐵 𝐵 such that 𝑑a(𝑥) = [𝑎, 𝑥] for all 𝑥 ∈ 𝑅.

Then it can be shown that 𝑑𝑎 is a derivation of 𝑅. This 𝑑𝑎 is called an inner

derivation of 𝑅 determined by `𝑎′ and usually it is denoted by 𝐼𝑎 .Clearly,

every inner derivation of a ring 𝑅 is a derivation. But the converse needs not

be true in general.

For example, take the ring 𝐵 =

0 𝛼 𝛽

0 0 𝛾

0 0 0

𝛼, 𝛽, 𝛾 ∈ 𝑍 with respect to

matrix addition and matrix multiplication. Define a map 𝑑: 𝐵 → 𝐵 by

8

𝑑

0 𝛼 𝛽

0 0 𝛾

0 0 0

=

0 𝛼 0

0 0 −𝛾

0 0 0

.

Remark 1.2

(i). It is easy to check that 𝑑 is a derivation of 𝐵, which is not an inner derivation

of 𝐵.

(ii). An additive mapping 𝑑: 𝐵 → 𝐵 is called a Jordan derivation if

𝑑 x2 = 𝑥𝑑(𝑥) + 𝑥𝑑(𝑥) for all 𝑥 ∈ 𝐵.

(iii). Every derivation is a Jordan derivation but the converse needs not be true in

general.

c. Let 𝑁𝑅 be a non empty set equipped with two binary operations say „+’ and

„∙‟, then 𝑁𝑅 is called a left near ring ( respectively a right near-ring) if:

(i). (𝑁𝑅 , +) is a group (not necessarily abelian),

(ii). (𝑁𝑅 , ∙) is a semi group,(i.e. 𝑁𝑅satisfies closure and assotiative law

with regards to multiplication),

(iii). 𝑥 𝑦 + 𝑧 = 𝑥 𝑦 + 𝑥 𝑧 ∀ 𝑥, 𝑦, 𝑧 𝑁𝑅 (respectively,

𝑦 + 𝑧 𝑥 = 𝑦 𝑥 + 𝑧𝑥 , ∀ 𝑥, 𝑦, 𝑧 𝑁𝑅).

The examples of such structure are the following:

(i). Let (ℂ,+) be the usual group of complex numbers with respect to ordinary

addition of complex numbers. Let us define `∗’ in ℂ as 𝑥 ∗ 𝑦 = x yfor all

∀ x, y ∈ ℂ.Then (ℂ,+,∗) is a left near ring which is not a right near ring.

(ii). Let (𝐺, +) be a non abelian group. Consider 𝑆, the set of all functions

9

From𝐺 to 𝐺. Then (𝑆,+,∙) is a right near-ring, but not a left near-ring,

with regard to the operation of addition `+’ and multiplication `∙’ defined as:

(𝑓 + 𝑔)(𝑥) = 𝑓(𝑥) + 𝑔(𝑥) ∀ 𝑥∈𝐺, and

(𝑓 𝑔) (𝑥) = 𝑓 (𝑔(𝑥))∀ 𝑥∈𝐺,where𝑓,𝑔∈𝑆.

A left near- ring 𝑁𝑅 is called zero symmetric if 0x = 0 for all 𝑥∈𝑁𝑅.

Throughout the discussion, 𝑁𝑅 will denote a zero symmetric left near-ring

with center 𝑍 (𝑁𝑅) unless otherwise mentioned.

d. Let 𝑁𝑅 be a near-ring. Then

(i). An additive mapping 𝑓∶ 𝑁𝑅→𝑁𝑅 is called a right generalized

derivation of 𝑁𝑅 if there exists a derivation d of 𝑁𝑅 such that

𝑓(𝑥𝑦) = 𝑓(𝑥)𝑦 + 𝑥𝑑(𝑦) 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥,𝑦 ∈ 𝑁𝑅.

(ii). An additive mapping 𝑓∶ 𝑁𝑅→𝑁𝑅 is called a left generalized

derivation of 𝑁𝑅 if there exists a derivation 𝑑 of 𝑁𝑅 such that

𝑓(𝑥𝑦) = 𝑑(𝑥)𝑦 + 𝑥𝑓(𝑦) 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥,𝑦 ∈𝑁𝑅.

(iii). An additive mapping 𝑓∶ 𝑁𝑅→𝑁𝑅 is called a generalized derivation

of𝑁𝑅 if ∃ a derivation 𝑑 of𝑁𝑅such that 𝑓(𝑥𝑦) = 𝑓(𝑥)𝑦 + 𝑥𝑑(𝑦) for

all 𝑥,𝑦 ∈𝑁𝑅 and 𝑓(𝑥𝑦) = 𝑑(𝑥)𝑦 + 𝑥𝑓(𝑦) for all 𝑥,𝑦 ∈𝑁𝑅.

For example, let 𝑆 be any zero symmetric left near-ring. Take

𝑆1 = 0𝛼0𝛽 0,𝛼,𝛽∈𝑆 . Then 𝑆1 is a zero symmetric left near- ring with regard to the matrix addition and multiplication.

10

e. Define 𝐹: 𝑆1→𝑆1and 𝐷: 𝑆1→𝑆1 by 𝐹 0𝛼0𝛽 = 000𝛽 and

𝐷 0𝛼0𝛽 = 0𝛼00 . It can be checked that 𝐹 is a right generalized

derivation of 𝑆1 with associated derivation 𝐷 of 𝑆1 but it is not a left

generalized derivation of 𝑆1with associated derivation 𝐷 of 𝑆1.

For example, consider 𝑆2= α𝛽00 0,𝛼,𝛽∈𝑆 .Then𝑆2 is a zero symmetric left near-ring with respect to the matrix addition and multiplication.

f. Define 𝐹: 𝑆2→𝑆2and 𝐷: 𝑆2→𝑆2 by 𝐹 𝛼𝛽00 = 0𝛽00 and

𝐷 0𝛼0𝛽 = 0𝛼00 .

It can be easily seen that 𝐹 is a left generalized derivation of 𝑆2 with

associated derivation 𝐷 of 𝑆2but 𝐹 is not a right generalized derivation of

𝑆2with associated derivation 𝐷 of 𝑆2.

For example,consider 𝑆3= 0𝛼𝛽00000𝛾 0,𝛼,𝛽,𝛾∈𝑆 . Then 𝑆3 is a zero symmetric left near-ring with regard to the matrix addition and multiplication.

11

g. Define 𝐹: 𝑆3→𝑆3and 𝐷: 𝑆3→𝑆3 as F 0𝛼𝛽00000𝛾 = 000000000 and

𝐷 0𝛼𝛽00000𝛾 = 0𝛼0000000 .Then it can be easily seen that F is a

generalized derivation of 𝑆3with associated derivation 𝐷 of 𝑆3.

Subsequently, we replace 𝑁𝑅 by 𝑁 to denote near-ring for simplicity.

1.7 Organization of the Thesis

This thesis is divided into six chapters. Chapter one covers the general introduction of the thesis and Chapter two dicusses the review of selected literature. The fundamentals of commutative theorems for rings are explicated in Chapter three while derivations of some special classes of 𝜎− rings are covered in Chapter four. Chapter five gives the results of some special classes of near-rings. Chapter six summarises, concludes and gives direction for future research. 1.8 Limitation of the study.

This research work is limited to the study of commutativity theorems for semi prime ring, ring with unity 1, s-unital rings and n-(s, t) permutting derivations on prime near-rings. It studies how non commuatative rings mentioned above under suitable constraints become commutative rings. In addition the notion of n-(s, t) permutting derivations on prime near-rings has been generalized to any positive integer n but to work it out for larger n (n ≥ 10), one needs a computer.