A Study on Commutativity Theorems for Rings and Near- Rings
Chapter One
AIMS & 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 derivation as well as permuting derivations on prime near-rings.
In order to achieve the above aim, the objectives considered are to:
- extend the related results for one sided s-unital rings and n-torsion free rings, (ii). establish the resultsof Jordan right derivation and generalized Jordan right derivation on rings,
- introduce the notion of permuting4 derivation as well as permuting derivation in near-rings,
- show that additive commutativity of a near ring satisfies certain identities involving permuting derivations of a prime near ring,
- give examples to justify the notions of permuting4 − derivation and permuting derivations,
- extend Posner‟s first theorem to prime rings of characteristic different from two,
- examine polynomial identities with constraints such as commutators, torsion free conditions on rings and near-rings thereby establishing commutativity theorems.
CHAPTER TWO
LITERATURE REVIEW
The origin of commutativity theorems for rings could be traced to the work of Wedderburn (1905) titled “A finite division ring is necessarily a field” in Transaction American Mathematical society. This result has attracted the attention of most mathematicians because it was so unexpected,interrelating two seemingly unrelated things, namely the number of elements in certain algebraic systems and the multiplication of the system.For algebraists, the mentioned Wedderburn theorem served as a jumping-off point for a large area of research in commutativity of rings in the 1950s.
Jacobson (1945)proved that Algebraic division algebra over a finite field is commutative.During the last several decades, there have been many results concerning conditions that force a ring to be commutative. There are now more than 400 papers in which conditions are given that determine commutativity for a ring or for special type of rings such as prime rings, semi-prime rings, rings with identity 1 and s-unital rings. These results were stimulated by the famous result ofJacobson (1964) which stated that if for every x in a ring R there exists a positive integer
n(x) such that xn(x) = x , then R is commutative.
Kaplansky (1948) proved that if a division ring satisfies any polynomial identity then it is finite dimensional over its center. Kaplansky (1951) asserted that, if R be is a ring with center ) and a positive integer > 1 is such that
∈ for every , then If in addition is semi simple then is commutative.
Herstein (1953)proved that if a rin satisfying the property that for every there exists an integer > 1 such that − ,then is commutative. Faith (1960) showed that, is commutative.
CHAPTER THREE
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FUNDAMENTALS OF COMMUTATIVITY THEOREMS FOR RINGS
We know that in a commutative ring , the commutators are central. It is natural and interesting to question whether a ring in which all the commutators are central, needs be commutative? In general, the commutators need not be centralized. The ring of 3 × 3 strictly upper triangular matrices over a field F is one of the examples of those rings which satisfy the condition but it is not commutative. It is, rather, surprising that this problem could not be investigated till 1962 when Herstein
proved that a division ring in which is central for every pair of must be commutative. We discussthe techniques involve in commutativity of a semi prime ring.
CHAPTER FOUR
DERIVATIONS ON SOME SPECIAL CLASS OF RINGS
Definitions of Some Rings and Derivations Definition1.1
An additive mapping on a ring is called an involution if and hold for all . A ring equipped with an involution is called a ring with involution or ring.
CHAPTER FIVE
RESULTS ON SOME SPECIAL CLASSES OF NEAR-RINGS
Near-ring Theoretic Concepts
A long standing result due to Herstein (1968) asserts that a periodic ring is commutative if its nilpotent elements are central.Ligh (1989) has raised a question whether similar result willhold for distributively generated (d-g) near-rings. Answering that, Bell (1973), gave an affirmative answer and proved that if N is a (d-near- ring with its nilpotent elements lying in the center,and that the set I of nilpotent elements forms an ideal and N/I is periodic, then N must be commutative. Herstein (1968) asserts that if a ring R satisfies the identity [xn, y] = [x, y], then R is commutative.
CHAPTER SIX
SUMMARY, CONCLUSION AND RECOMMENDATIONS
Summary
This thesis presents some contributions to commutativity theorems for rings, near- rings, and applications in permuting derivation in near-rings
We first discuss the general introduction of the thesis, which includes the historical background of the study, motivation and justification, and the aim and objectives of the study. Then a comprehensive and critical literature survey of the fundamentals of commutativity of rings and near-rings is presented.
We then establish the extension of commutativity condition, and then prove that certain classes of rings such as semi prime rings, when the commutators are central, with condition given below:
[(xno ym)k ± (xmo y), x] = 0 or [(xn o ym)k ± (xm o y), y] = 0 for all ring elements and fixed natural numbers
Next a ring theoretic analogue of group theoretic results which asserts that a group is commutative if and only if for all in G is presented.We then studyother polynomial identities and investigate the classes of commutative rings satisfyinggeneralized forms of these identities. We obtain a result which states that if R is a ring with 1 satisfying where for every there exist polynomials Î and are fixed positive integers.Also with
positive integer and together with at least one of is zero, then R is a commutative ring. Also some results onrings with unity and s- unital rings are presented.
We then present our results which were derived after relaxation of Khan (2001), Putcha and Yaqub (1973) that a ring satisfying a polynomial identity of the form xy = w(x,y), where w (x, y) (X, Y) is a word different from XY in non commuting in determinates X and Y, must have a commutator ideal.
Furthermore, we introduce the new notion of permuting n -(s, t) derivations on prime near rings and establishes related results.
Finally we prove the Posner‟s first theorem in to s -prime ring Jordan left s ring ( respectively Jordan right s ring).The generalizations of some ofthe results presented in this thesis may be possible, but the choice of our examples show that they cannot be extended arbitrarily.
Conclusion
In this thesis, some results on commutativity of semi prime rings, ring with unity, s- unital rings and permuting 4 − derivation as well as permuting derivations on prime near-rings studied. We obtained some related results for one sided s-unital rings and n-torsion free rings, established the results of Jordan right−derivation and generalized Jordan righ−derivation on − rings and showed that additive commutativity of a near ring satified certain identities involving permuting derivations of a prime near ring. We extended Posner‟s first theorem to −prime rings of characteristic different from two and examined polynomial identities with constraints such as commutators, torsion free conditions on rings and near-rings thereby establish commutativity theorems.
Recommendations
The notion provided in subsection 5.2.1 and 5.2.2 may be found useful in addressing some real life problems which require symmetry of abstraction. The concept of semiprime rings, rings with identity and prime near-rings can be very useful in many areas like coding theory, cryptography and decision making, etc. In view of the derivation application of rings and near-rings,it needs to be investigated whether or not, similar to the result that every semiprime rings can be extended into arbitrary rings found in (Khan et al., 2013) also holds for s-unital rings as well. One can investigate the possibility of the commutativity of addition and multiplication of near- rings satisfying some algebraic or differential identities involving one of the properties, permuting -derivations, permuting generalised derivations, permuting derivation andpermuting generalised derivationon semi group ideals of near-rings.
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