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
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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
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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.
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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
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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
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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.
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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.
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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.
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