GENERALIZED HIGHER LEFT CENTRALIZER OF PRIME RINGS

Throughout this paper R is a ring, R is called prime if aRb=(0) implies a=0 or b=0 with a,b aRa=(0) with aR implies a=0 R is called 2 denotes the commutater ab-ba. An additive mapping d of R int +ad(b) (resp. d(a)= d(a)a+ad(a)), for all a,b mapping f of R into itself is called generalized derivation ( Jordan derivation) d of R such that f(ab) = f(a)b + ad(b) was extended to higher derivation by Hasse and Schmidt family of additive mappings of R into itself then D is


INTRODUCTION
Throughout this paper R is a ring, R is called prime if aRb=(0) implies a=0 or b=0 with a,b aRa=(0) with aR implies a=0 R is called 2 denotes the commutater ab-ba.An additive mapping d of R int +ad(b) (resp.d(a 2 )= d(a)a+ad(a)), for all a,b mapping f of R into itself is called generalized derivation ( Jordan derivation) d of R such that f(ab) = f(a)b + ad(b) was extended to higher derivation by Hasse and Schmidt family of additive mappings of R into itself then D is ) hold for all a,b Haetinger (2000) proved that on a prime ring with 2 Haetinger (2005) defined a generalized higher derivations as follows, let F=(f called generalized higher derivation (resp.generalized Jordan higher derivation) if there exists higher derivation(resp.Jor higher derivation) D=(d i ) i  N such that n ab f ( right non-zero divisor and U a square closed Lie ideal of R then every generalized Jordan higher derivation is generalized higher derivation.Following Zalar (1991), an additive mapping T from R T(a)b (resp.T(ab) =aT(b)) holds for all a,b proved that any left (resp.right) Jordan centralizer on 2 In this paper we present the concepts of higher left centralizer, Jourdan higher left centralizer also we introduce t generalized higher left centralizer, generalized Jordan higher left centralizer of R, as well as we prove that every Jordan left centralizer of 2-torsion free prime ring R is higher left centralizer of R. Also we prove every generali centralizer of 2-torsion free prime ring R is generalized higher left centralizer of R.
Throughout this paper R is a ring, R is called prime if aRb=(0) implies a=0 or b=0 with a,b R implies a=0 R is called 2-torsion free 2a=0 then a=0 with aR (Vokman and Ulbl ba.An additive mapping d of R into itself is called derivation (resp.Jordan derivation) if d(ab) = d(a)b  Havala (1998) presented the concept of generalized derivation as follow an additive called generalized derivation (resp.generalized Jordan derivation) if there exists a derivation f(ab) = f(a)b + ad(b) (resp.f(a 2 ) = f(a)a + ad(a)) hold for all a,b was extended to higher derivation by Hasse and Schmidt (1937) also see (Vokman and Ulbl, 2003 family of additive mappings of R into itself then D is called higher derivation (resp.Jordan higher derivation) of R if ) hold for all a,b R. In an attempt to generalize Herstein result for higher derivations, proved that on a prime ring with 2-torsion free every Jordan higher derivation is a higher derivation.Cortes and defined a generalized higher derivations as follows, let F=(f i ) i N be a family of additive mapping of R then F is called generalized higher derivation (resp.generalized Jordan higher derivation) if there exists higher derivation(resp.Jor and they prove that if R is 2-torsion free ring which has commutator zero divisor and U a square closed Lie ideal of R then every generalized Jordan higher derivation is generalized higher , an additive mapping T from R into itself is called a left (resp.right R and R is called semiprime if Vokman and Ulbl, 2003) as usual [a,b] resp.Jordan derivation) if d(ab) = d(a)b presented the concept of generalized derivation as follow an additive resp.generalized Jordan derivation) if there exists a derivation (resp. ) = f(a)a + ad(a)) hold for all a,b R. The concept of derivation 2003) as follow, let D=(d n ) n  N be a resp.Jordan higher derivation) of R if Herstein result for higher derivations, torsion free every Jordan higher derivation is a higher derivation.Cortes and be a family of additive mapping of R then F is called generalized higher derivation (resp.generalized Jordan higher derivation) if there exists higher derivation(resp.Jordan torsion free ring which has commutator zero divisor and U a square closed Lie ideal of R then every generalized Jordan higher derivation is generalized higher into itself is called a left (resp.right) centralizer of R if T(ab)= R, if T is both left as well as right centralizer, then its called a centralizer.B. Zalar tortion free semiprime ring is a left (resp.right) centralizer.
In this paper we present the concepts of higher left centralizer, Jourdan higher left centralizer also we introduce the concepts of generalized higher left centralizer, generalized Jordan higher left centralizer of R, as well as we prove that every Jordan higher torsion free prime ring R is higher left centralizer of R. Also we prove every generalized Jordan higher left

Jordan Higher Left Centralizer
In this section we present the concepts of higher left centralizer, Jordan higher left centralizer of a ring R also we study the properties of them.We begin with the following definition: Definition 2.1: Let T= (t i ) i N be a family of additive mappings of a ring R into itself.Then T is called higher left centralizer if for every nN for all a,bR …(1) And T is called Jordan higher left centralizer of R if for every nN T is called Jordan triple higher left centralizer of R if for every nN for all a,bR …(3) Lemma 1: Let T= (t i ) i N be Jordan higher left centralizer of a ring R into itself then for all a,b,cR, nN On the other hand Comparing ( 1) and (2) we get On the other hand Comparing ( 1) and (2) we get 3) Since R is commutative and from (2) we get Since R is 2-torsion free we get the require result.
Definition 2.2: Let T=( t i ) i N be a family of higher Jordan left centralizer of a ring R and nN  Lemma 2: Let T=( t i ) i N be a family of higher Jordan left centralizer of a ring R then for all a,b,c  R and nN 1) As the same way of ( 1) Lemma 3: Let T=( t i ) i N be a family of higher Jordan left centralizer of 2-torsion free prime ring R then for all a,b R and nN On the other hand Compare (1) and (2) we get  0= (t(ab) -t(a) b)m On the other hand Compare ( 1) and (2) we get Now, replace b+d for b in Lemma 3, we get By (1) and (2) and Lemma 3, we get Theorem 5: Every Jordan higher left centralizer of 2-torsion free prime ring R is higher left centralizer of R.
Proof: Let T= ( t i ) i N be Jordan higher left centralizer of prime ring R.
Since R is prime, we get from Theorem 4 , either 0 for all c,dR, nN then R is commutative ring and by Lemma 1 we get Since R is 2-torsion free, we obtain T is a higher left centralizer of R.
Proposition 6: Let T= ( t i ) i N be Jordan higher left centralizer of 2-torsion free ring R, then T is Jordan triple higher left centralizer of R.
Proof : Replace b by ab + ba in Definition 2.1, then : On the other hand : t n (a(ab+ba) + (ab+ba)a) = t n (aab + aba + aba+ baa) = t n (aab + baa) + 2t n (aba) Now, compare (1) and (2) we get Since R is 2-torsion free we obtain that T is Jordan triple higher left centralizer of R.

3) Generalized higher left centralizer of rings
In this section we present the concepts of generalized higher left centralizer and generalized Jordan higher left centralizer of rings also we present some properties of them.Definition 3.1: Let F= (f i ) i N be a family of additive mappings of a ring R into itself.F is called generalized higher left centralizer of R if there exists a higher left centralizer T=(t i ) i N of R such that for every nN we have for all a,bR where T is called the relating higher left centralizer.F is called Jordan generalized higher left centralizer of R if there exists a Jordan higher left centralizer of R if for every nN for all aR where T is called the relating Jordan left centralizer.
F is called Jordan generalized triple higher left centralizer of R if there exists a Jordan triple higher left centralizer of R if for every nN for all a,bR where T is called the relating Jordan triple left centralizer, Lemma 7: Let F= (f i ) i N be Jordan generalized higher left centralizer of a ring R into itself then for all a,b,cR , nN On the other hand Comparing ( 1) and ( 2) we get 2) Replace a+c for a in Definition 2.1 (iii) On the other hand Comparing ( 1) and ( 2) we get 3) Since R is 2-torsion free commutative ring and from (2) we get the require result.Definition 3.2: Let F=( f i ) i N be a family of Jordan generalized higher left centralizer of a ring R with relating Jordan higher left centralizer T=(t i ) i N of R , then for all a,bR and nN,we define Lemma 8: Let F=( f i ) i N be a family of higher Jordan left centralizer of a ring R with relating Jordan higher left centralizer T=(t i ) i N of R then for all a,b,c  R and nN 1) 2) As the same way of (1) 3) By Lemma 7 (i) On the other hand f n (w)= f n ( (ab)m(ba) + (ba)m(ab)) Compare ( 1) and (2) we get 0= Now, replace b+d for b in Lemma 9, we get Since R is 2-torsion free, we obtain T is a higher left centralizer of R.
Proposition 12: Let F= ( f i ) i N be Jordan generalized higher left centralizer of 2-torsion free ring R, then T is Jordan generalized triple higher left centralizer of R.

of Current Research, Vol. 08, Issue, 11, pp.40966-40975, November, 2016
Every Jordan generalized higher left centralizer of 2-torsion free prime ring R is generalized higher left centralizer of R.Proof: Let F= ( f i ) i N be Jordan generalized higher left centralizer of prime ring R. Since R is prime, we get from Theorem 10 , either n-1 (c ) ,t n-1 (d)]=0 for all a,b,c,dR and nN.If [t n-1 (c ) ,t n-1 (d)]  0 for all c,dR, nN then 2.3 we get T is higher left centralizer of R.If [t n-1 (c ) ,t n-1 (d)]=0 for all c,dR, nN then R is commutative ring and by Lemma 7 we get n or [t n by Remark