Sivakumar Work Correction

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Contents

Introduction

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1 NEAR RINGS 4 1.1 Basic Denitions and Examples . . . . . . . . . . 4 1.2 Ordered near rings . . . . . . . . . . . . . . . . . 11 2 N-GROUPS 25 2.1 Semisimple N-groups . . . . . . . . . . . . . . . . 25 2.2 Chain conditions an N-groups . . . . . . . . . . . 41 3 Generalized derivations an near rings 49 3.1 - derivations . . . . . . . . . . . . . . . . . . . 49 3.2 Generalized derivations . . . . . . . . . . . . . . . 54 Bibliography

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IntroductionRing theory is a show piece of mathematical unication, bringing together several branches of the subject and creating a powerful machine for the study of problems of considerable historical and mathematical importance. Near - rings are generalized rings. Near - rings arise in a natural way; take the set M () of all mapping of a group (, +) into itself, dene addition (+) pointwise and o as composition. Then (M (), +, ) is a near ring. Even if is a abelian, only one distributive law is always fullled. (f + g) h = f h + g h holds by the denition of f + g, while for f (g + h) = f g + f h we would have to assume that f is a homomorphism. Another example is supplied by the polynomials w.r.t addition and substitution. Near - ring provide non-linear theory of group mappings historically, the rst step towards near - rings was an axiomatic research is done by Dickson in 1905. He showed that there do exist elds with only one distributive law. Some years later these eld theory proved be useful in coordinating certain importance classes of geometric planes (Descartes plane and pauppian planes). A part from the applications concerning axiomatics and geometry mentioned above, the special classes of nite nearrings give new and highly ecient classes of balanced in complete block designs the characterize Frobenius group and hence

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also nite groups with xed point free automorphism groups. The notation of the ring with derivation is quite and plays a signicant role in the integration of analysis, algebraic geometry and algebra. In 1940s it was found that the Galois theory of algebraic equations can be transferred to the theory of ordinary dierential equations. The study of derivations in rings through initiated long back, but got impetus only after posner who in 1957 established two very striking results on derivations in prime rings. The notion derivation in rings has also been generalised in various directions such as generalized derivation, Jorchon derivations, generalized Jorden derivation also there has been considerable intergest in investigating commutativity of rings more often that of prime and semi prime rings. Analogus to be concept of derivation of rings, so on the concept of derivation on near - rings was intiated by H.E. Bell and G. Mason in 1987 [BM]. Since then only a few paper appeared in this topic [A], [BA], [H]. In 1991 Bresar. M introduced the notation of generalized derivations corresponding to the derivation on a ring R [B]. In [OG] generalized derivation of Prime near rings is discussed. This motivated as to study the concepts of near rings and generalized derivation of prime near rings. Our work is divided into 3 Chapters. The rst chapter on Near - Rings is divided into two sections the rst section deals with denitions and examples and the second section deals with Near - ring.2

The Second Chapter on -groups is divided into two sections the rst section deals with semisimple N- groups and the second section deals with chain conditions on N -groups. The third chapter discuses Generalized derivations on near rings. This chapters divided into two sections. The rst section deals with derivations and the second section deals with generalized derivations. Our work ends with a detailed bibliography.

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Chapter 1 NEAR RINGS1.1 Basic Denitions and Examples

Denition 1.1.1 A near ring N is a set N together with two binary operations, addition and multiplication, such that i (N, +) is a group (not necessarily abelian) ii (N, .) is a semigroup (not necessarily with an identity element) and iii For all x, y, z in N , (x + y)z = xz + yz (Right distributive law If the above conditions are satised for all elements in N , then N is called a right near ring. Similarly we can dene a left near ring. Through out our work we will consider only right near rings. Example 1.1.2 Let (G, +) be any group. Dene a multiplication on G as follows for all x, y in G , x y = x.4

Then (G, +, ) is a near ring. Proof Given that (G, +) is a group. we have to check the remaining conditions i. (G, ) is a semigroup. ii. If x1 , x2 , x3 G then (x1 + x2 ) x3 = x1 x3 + x2 x3 For x, y G, given that xy =x is colosed. claim (x y)z = x (yz) L.H.S = (x y) z = xz = x R.H.S = x (y z) = xy = x From(1) and (2) Therefore, (x.y).z = x.(y.z) x, y, z G5

(1)

(2)

in G is associative. Therefore (G, ) is a semigroup. Let x, y, z G. Then x z = x and y z = y x z + y x = x + y. Also (x + y) z = x + y by denition of . Hence (x + y).z = xz + yz Therefore (G, +, ) is a near ring.

Example 1.1.3 Let N = R[x], the set of all polynomials in x over the eld of real numbers R. Dene addition + and composition on N as follows: For p(x), q(x) N , let p(x) = a0 + a1 x + a2 x2 + . . . q(x) = b0 + b1 x + b2 x2 + . . . Then p(x) + q(x) = (a0 + b0 ) + (a1 + b1 )x + (a2 + b2 )x2 + . . . Denition 1.1.4 Near ring with identity Let N be a near ring, N is said to be near ring with identity, if there is an element 1 N such that 1.n = n.1 = nn N Example 1.1.5 The set N of all polynomials over the eld of real numbers, with constant term o togethers, with constant term6

o together with the usual addition and composition of polynoomials, is a near ring. Since (N, .) is a semigroup with an identity element 1, we call N a near ring with an identity element 1. Denition 1.1.6 Let N be a near ring and let S be a nonempty subset of N . Then S is said to be subnear ring of N , if S itselt is a near ring under the operations in N . Example 1.1.7 Set of continuous function from R R forms a subnear ring under usual addition and compsition of mapping Denition 1.1.8 Constant element Let N be a near ring. Let k N . Then k is said to be constant in N if kx = k N Denition 1.1.9 A near ring N is said to be constant near ring if every element of N is constant. Example 1.1.10 Example 1.1.2, (G, +) is group and . is dened as xy = x x, y G clearly each element of G is constant in G. Therefore (G, +, .) is a constant near ring. Example 1.1.11 Consider example 1.1.3, we observe that 0.p(x) = 0 p(x) IR[x] = N , Therefore 0(zero) is ther only constant term of N . x

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Lemma 1.1.12 An element x in N is a constant if and only if x = x0 proof Assume that x in N is a constant. claim x = x0 X is constant xy = xy N Equation (1) is true for all y N Therefore equation (1) is true for y=0 Therefore x0 = x which is required. Conversely assume that x = x0 claim x is a constant in N ie., we have to prove xy = x Given x = x0 For any y N . xy = (x0)y = x(0y) Associative law holds inN = x0 = x xy = x 0is the constant[0y = 0] using (2) y N which is required y N (2) (1)

Lemma 1.1.13 Let N be a near ring. Let k(N ) be the collection of all constant element in N . Then K(N ) is a subnear ring8

of N Proof K(N ) = {k N/ k is constant inN } x N }

= {k N/kx = k claim k(N ) is a subnear ring of N we have to prove (i) (k(N ), +) is a group (ii) (k(N ), .) is a semigroup (iii) ((x + y).z = xz + yz tributive law) (k(N ), +) is a group of N . Therefore 0.x = 0 x.z = x, y.z = y N is near ring, x.z y.z = x y N

x, y, z K(N ) (Right dis-

x N, 0 k(N ) k(N ) = Therefore z N

ie.x y = (x.z y.z) = (x y).z for f orx, y k(N ), x y k(N ) (k(N ), +) is a group. To prove (k(N ), .) is a semi group ie., we have to prove9

(i) If x, y k(N ) then x.y k(N ) (ii) x(yz) = (xy)z (i) Let x, y k(N ) claim xy k(N ) (xy)z = xyz N (x.y)z = x.(y.z) = x.yz N ie.,(xy.z = xyz N x.y k(N ), x, y, z k(N )

. in N is associative,and k(N ) N , . is associative in k(N ) Since each element of k(N ) is an element of N . associative lawa is true for k(N ). (k(N ), .) is a semigroup. Right distributive law Let x, y, z k(N ) claim (x + y).z = xz + yz L.H.S = (x + y).z = (x + y)10

(1)

x + y k(N ) R.H.S xz + yz = x + y (2)

From, (1) and (2), L.H.S = R.H.S (x + y).z = xz + yz distributive law is true (k(N ), +, .) is a subnear of N , since each element of k(N ) is constant. (k(N ), +, .) is a constant subnear ring of N Denition 1.1.14 Let N be a near ring. Let k(N ) be the collection of all constant elements in N . The subnearring of N . k(N ) is called the constant subnear ring

1.2

Ordered near rings

Denition 1.2.1 Linear order (or) simple order (or) order relation. Let N be a any non-empty set. Let C be any relation on N . Then C is said to be order relation if the following conditions are true, (i) Comparability For each x, y N for which x = y; either xCy or yCx11

(ii) Non reexivity There is no x in N such that xCx holds (iii) Transitivity If xCy and yCz then xCz, x, y, z N Example 1.2.2 let R be a real number system, x hold. If O < x, x p P is non-empty If O > x, O < x x p P is non-empty Hence (i) is proved. To Prove (ii) If a p , then for all x in N x+a+x p.13

Let a p. Then a > 0 Let x N a > 0 x + a > x + 0 (x + a) + x > x + x x + a + x > 0 x + a + x p

Hence (ii) is proved To Prove (iii) Let a, b P, a > 0 and b > 0, a > 0 a + b > 0 + b linear order a+b>b>0 a+b>0a+bp a>0 ab > 0.b ab > 0 ab p

Hence (iii) is proved. To Prove (iv) Let x N we know that N = 0 There is an element x N such that x = 0. since N is an14

ordered near ring, only one of the follo