# Annihilator condition of a pair of derivations in prime and semiprime rings

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ANNIHILATOR CONDITION OF A PAIR OF DERIVATIONS IN PRIME AND SEMIPRIME RINGS1

Basudeb Dhara, Nurcan Argac¸∗∗and Krishna Gopal Pradhan

Department of Mathematics, Belda College, Belda, Paschim Medinipur 721 424,

West Bengal, India

∗∗Department of Mathematics, Science Faculty, Ege University, 35100,

Bornova, Izmir, Turkey

e-mails: basu dhara@yahoo.com; nurcan.argac@ege.edu.tr; kgp.math@gmail.com

(Received 10 December 2013; after final revision 29 October 2015;

accepted 2 November 2015)

Letn be a fixed positive integer, R be a prime ring,D andGtwo derivations of R andLa

noncentral Lie ideal ofR. Suppose that there exists06=a∈Rsuch thata(D(u)ununG(u)) = 0for allu∈L, wheren≥1is a fixed integer. Then one of the following holds:

1. D=G= 0, unlessRsatisfiess4;

2. char(R)6= 2,Rsatisfiess4,nis even andD=G;

3. char(R)6= 2,Rsatisfiess4,nis odd andDandGare two inner derivations induced by

b, crespectively such thatb+c∈C;

4. char(R) = 2andRsatisfiess4.

We also investigate the case whenRis a semiprime ring.

Key words : Prime ring; derivation; extended centroid; Martindale quotient ring.

1. INTRODUCTION

Throughout this paper,R always denotes a prime ring with centerZ(R), extended centroidC and Q is its Martindale quotient ring. Forx, y R, the Lie commutator of x, y is denoted by [x, y]

1

This work is supported by a grant from National Board for Higher Mathematics (NBHM), India. Grant No. is

NBHM/R.P. 26/ 2012/Fresh/1745 dated 15.11.12 and second author is supported by The Scientific and Technological

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and defined by[x, y] = xy −yx. An additive subgroupL ofR is said to be a Lie ideal ofR, if [L, R] L. ByDwe mean a derivation ofR, that is an additive mappingD : R R satisfying D(xy) = D(x)y+xD(y)for allx, y ∈R. A derivationDis calledQ-inner if it is inner induced

by an element, sayq ∈Qas an adjoint, that is,D(x) = [q, x]for allx R. A derivation which is

notQ-inner is called a Q-outer derivation. The standard polynomial identitys4 in four variables is

defined ass4(x1, x2, x3, x4) =

P

σ∈S4

(1)σx

σ(1)(2)(3)(4)where(1)σ is+1or1according

toσbeing an even or odd permutation in symmetric groupS4.

A well known result proved by Posner , states that if the commutators[D(x), x] Z(R)

for all x R, then either D = 0 or R is commutative. Then many related generalizations of

Posners result have been obtained by a number of authors in literature. Breˇsar proved in  that if D(x)x−xG(x) Z(R)for allx R, then eitherD = G = 0orRis commutative. Later Lee

and Wong  studied the same situation of Breˇsar for allxin some noncentral Lie idealLofRand

obtained that eitherD=G= 0orRsatisfiess4.

Recently, Argac¸ and De Filippis  obtained the following result:

LetRbe a prime ring with char(R)6= 2,La non-central Lie ideal ofR,D, Gtwo derivations of Randn≥1a fixed integer. If(D(x)x−xG(x))n= 0for allxL, then eitherD=G= 0orR

satisfies the standard identitys4andD, Gare inner derivations, induced respectively by the elements

aandbsuch thata+b∈Z(R).

Recently, first author of this paper  studied the case with left annihilator condition. More

precisely, he obtained the following result:

LetRbe a prime ring,DandGtwo derivations ofRandLa noncentral Lie ideal ofR. Suppose

that there exists06=a∈Rsuch thata(D(u)u−uG(u))n= 0for allu∈L, wheren≥1is a fixed

integer. Then one of the following holds:

(i)D=G= 0unlessRsatisfiess4;

(ii) char(R) = 2andRsatisfiess4;

(iii) char(R)6= 2andRsatisfiess4,DandGare two inner derivations induced byp, qrespectively

such thatp+q∈C.

In , Lee and Zhou studied the situation for generalized derivations as follows:

Let R be a prime ring that is not commutative and such that R 6∼= M2(GF(2)), let G, H be

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G(x) =xwandH(x) =wxfor allx∈R; (2) eitherw∈C, orxmandxnareC-dependent for all x∈R.

Carini and De Filippis  studied the situation in a noncentral Lie ideal of a prime ringR with

central values. More precisely, they proved the following theorem:

LetRbe a prime ring,U the Utumi quotient ring ofR,C = Z(U)the extended centroid ofR, La non-central Lie ideal ofR,HandGnon-zero generalized derivations ofR. Suppose that there

exists an integern≥ 1such thatH(un)un+unG(un) C, for allu ∈L, then either there exists a∈U such thatH(x) =xa, G(x) = −ax, orRsatisfies the standard identitys4. Moreover, in the

last case the structures of the mapsG, Hare obtained.

In , Chang et al. proved the following:

Letnbe a fixed positive integer. LetR be a noncommutative(n+ 1)!-torsion free prime ring.

Suppose that there exist Jordan derivationsDandG:R →Rsuch thatD(x)xn−xnG(x) = 0for

allx∈R. Then we haveD= 0andG= 0.

In the present paper, our aim is to study the same situation of derivations with left annihilator

condition in a prime ringRthat is,a(D(x)xn−xnG(x)) = 0for allx∈L, whereLis a noncentral

Lie ideal ofR.

LetQbe the Martindale quotient ring of a prime ringRandCthe center ofQ, which is called

ex-tended centroid ofR. Note thatQis also a prime ring withCa field. We denote byT =Q∗CC{X},

the free product overC of theC-algebraQand the freeC-algebraC{X}, withXthe countable set

consisting of the noncommuting indeterminatesx1, x2, . . .. The elements ofT are called generalized

polynomial with coefficients inQ. Nontrivial generalized polynomial means a nonzero element ofT.

For more details about these objects we refer to [3, 14].

2. THE RESULTS ON PRIMERINGS

First we fix the following remarks:

Remark 2.1 : Let R be a prime ring and La noncentral Lie ideal of R. If char(R) 6= 2, by

[4, Lemma 1] there exists a nonzero idealI ofR such that0 6= [I, R] L. If char(R) = 2and dimCRC > 4i.e., char(R) = 2andRdoes not satisfys4, then by [18, Theorem 13] there exists a

nonzero idealIofRsuch that06= [I, R]⊆L. Thus if either char(R)6= 2orRdoes not satisfys4,

then we may conclude that there exists a nonzero idealI ofRsuch that[I, I]⊆L.

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a derivation ofQ, and so any derivation ofRcan be defined on the whole ofQ. We refer to [3, 20]

for more details.

We begin with lemmas

Lemma 2.3 — Let nbe a fixed positive integer, R be a prime ring with extended centroid C

and a, b, c R. If a 6= 0 such that a(b[x1, x2]n+1 [x1, x2]b[x1, x2]n [x1, x2]nc[x1, x2] + [x1, x2]n+1c) = 0 for all x1, x2 R, where n 1 is a fixed integer, then either R satisfies a

nontrivial generalized polynomial identity (GPI) orb, c∈C.

PROOF : Assume that R does not satisfy any nontrivial GPI. If R is commutative, trivially R satisfies a nontrivial GPI which is a contradiction. So, R must be noncommutative. Let T = Q∗C C{X1, X2}, the free product ofQandC{X1, X2}, the freeC-algebra in noncommuting

in-determinates X1 and X2. Then, since a(b[x1, x2]n+1 [x1, x2]b[x1, x2]n [x1, x2]nc[x1, x2] + [x1, x2]n+1c) = 0is a GPI forR, we see that

a(b[X1, X2]n+1[X1, X2]b[X1, X2]n−[X1, X2]nc[X1, X2] + [X1, X2]n+1c) = 0 (1)

inT =Q∗CC{X1, X2}. Ifc /∈C, thencand1are linearly independent overC. Thus, (1) implies

a([X1, X2]n+1c) = 0

inT. This impliesc= 0, a contradiction. Hence we conclude thatc∈C. Then (1) reduces to

a(b[X1, X2][X1, X2]b)[X1, X2]n= 0 (2)

inT. Again by the same way, this implies thatb∈C. 2

Lemma 2.4 — Let n be a positive integer, R be a noncommutative prime ring with extended

centroidC anda, b, c R. Suppose that a = 06 such that a(b[x1, x2]n+1 [x1, x2]b[x1, x2]n [x1, x2]nc[x

1, x2] + [x1, x2]n+1c) = 0for allx1, x2 ∈R, wheren≥1is a fixed integer. Then one of the following holds:

1. b, c∈C, unlessRsatisfiess4;

2. char(R)6= 2,Rsatisfiess4,nis even andb−c∈C;

3. char(R)6= 2,Rsatisfiess4,nis odd andb+c∈C;

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PROOF: By assumption,Rsatisfies generalized polynomial identity

g(x1, x2) =a(b[x1, x2]n+1[x1, x2]b[x1, x2]n−[x1, x2]nc[x1, x2] + [x1, x2]n+1c). (3)

By Lemma 2.3, b, c C which gives the conclusion (i) unless R satisfies a nontrivial GPI.

Now, assume thatR satisfies a nontrivial GPI. SinceRandQsatisfy same generalized polynomial

identities (see ), Qsatisfiesg(x1, x2). Moreover, if C is infinite, we haveg(x1, x2) = 0for all

x1, x2 ∈Q⊗C C, whereCis the algebraic closure ofC. Since bothQandQ⊗C Care prime and

centrally closed , we may replaceRbyQorQ⊗C Caccording toC finite or infinite. Thus we

may assume thatC =Z(R)andRisC-algebra centrally closed, which satisfiesg(x1, x2) = 0. By

Martindale’s theorem ,Ris then a primitive ring and hence is isomorphic to a dense ring of linear

transformations of a vector spaceV overC.

Suppose that dimCV 3.

We shall show that for anyv ∈V,vandcvare linearlyC-dependent. Suppose thatvandcvare

linearly independent for somev V. Since dimCV 3, there existsu V such thatv, cv, uare

linearlyC-independent set of vectors. By density, there existx1, x2∈Rsuch that

x1v= 0, x1cv = 0, x1u=cv; x2v=cv, x2cv =u, x2u= 0.

Then[x1, x2]v= 0,[x1, x2]cv =cvand so0 =a(b[x1, x2]n+1[x1, x2]b[x1, x2]n−[x1, x2]nc[x1, x2]+ [x1, x2]n+1c)v=acv.

This implies that ifacv6= 0, by contradiction, we can say thatvandcvare linearlyC-dependent.

Now choosev∈V such thatvandcvare linearlyC-independent.

Then acv = 0. SetW = SpanC{v, cv}. Let ac 6= 0. Then there existsw V such that

acw 6= 0 and thenac(v−w) = −acw 6= 0. By the previous argument we have that w, cw are

linearlyC-dependent and(v−w), c(v−w)too. Thus there existα, β∈Csuch thatcw =αwand c(v−w) =β(v−w). Thencv=β(v−w) +cw=β(v−w) +αwi.e.,(α−β)w=cv−βv ∈W.

Nowα=βimplies thatcv=βv, a contradiction. Henceα6=βand sow∈W.

Again, ifu ∈V withacu= 0thenac(w+u)6= 0. So,w+u ∈W forcingu∈W. Thus it is

observed thatw∈V withacw 6= 0impliesw∈W andu ∈V withacu= 0impliesu ∈W. This

implies thatV =W i.e., dimCV = 2, a contradiction. Hence we conclude thatac= 0.

Again sincev, cv, uare linearlyC-independent set of vectors, by density, there existx1, x2 ∈R

such that

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Then [x1, x2]v = 0, [x1, x2]cv = v + cv and so 0 = a(b[x1, x2]n+1 [x1, x2]b[x1, x2]n

[x1, x2]nc[x

1, x2] + [x1, x2]n+1c)v =av. Sincea6= 0, by above argument this leads to a contra-diction.

Hence,v andcvare linearly C-dependent for allv V. Thus for each v V, cv = αvv for

someαv C. It is very easy to prove thatαv is independent of the choice ofv V. Thus we can

writecv=αvfor allv ∈V andα∈Cfixed. Now letr∈R,v∈V. Sincecv=αv,

[c, r]v= (cr)v−(rc)v=c(rv)−r(cv) =α(rv)−r(αv) = 0.

Thus[c, r]v = 0for allv∈V i.e.,[c, r]V = 0. Since[c, r]acts faithfully as a linear

transforma-tion on the vector spaceV,[c, r] = 0for allr ∈R. Therefore,c∈Z(R).

Therefore, from (3) we have thatRsatisfies generalized polynomial identity

g(x1, x2) =a[b,[x1, x2]][x1, x2]n= 0. (4)

Letvandbv are linearly independent overC. Since dimCV 3, there existsw ∈V such that

v, cv, ware linearlyC-independent set of vectors. By density, there existx1, x2 ∈Rsuch that

x1v= 0, x1bv=v, x1w=−v+ 2bv; x2v=bv, x2bv=w, x2w= 0.

Then[x1, x2]v =v,[x1, x2]bv=−v+bvand so0 =a[b,[x1, x2]][x1, x2]nv=av. Then again

by above arguments, sincea 6= 0, this leads a contradiction. Hence we conclude thatvandbv are

linearly dependent for allv∈V, implyingb∈C.

If dimCV = 2, thenR∼=M2(C), that is,Rsatisfiess4. If char(R) = 2, we get our conclusion

(4). So let char(R) 6= 2. Now we use the fact [x, y]2 Z(M2(C))for all x, y M2(C), and

consider the following two cases:

Letnbe an even integer. Then we have from (3) thatRsatisfies

a[x1, x2]n[b−c,[x1, x2]] = 0. (5)

By Lemma 2.1 in , since char(R)6= 2, we haveb−c∈C, which is our conclusion (2).

Letnbe an odd integer. Then we have from (3) thatRsatisfies

a[x1, x2]n[b+c,[x1, x2]] = 0. (6)

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Theorem 2.5 — Letnbe a fixed integer,Rbe a prime ring,DandGtwo derivations ofR,La

noncentral Lie ideal ofR. Suppose that there exists06=a∈Rsuch thata(D(u)un−unG(u)) = 0

for allu∈L, wheren≥1is a fixed integer. Then one of the following holds:

1. D=G= 0, unlessRsatisfiess4;

2. char(R)6= 2,Rsatisfiess4,nis even andD=G;

3. char(R) 6= 2,R satisfiess4,nis odd andDandGare two inner derivations induced byb, c

respectively such thatb+c∈C;

4. char(R) = 2andRsatisfiess4.

PROOF: If char (R) = 2andR satisfiess4, we obtain our conclusion (4). So, let either char (R)6= 2orRdoes not satisfys4. SinceLis a noncentral Lie ideal ofR, by Remark 1, there exists a

nonzero idealI ofRsuch that[I, I]⊆L. Hence, by our assumption we have,

a(D([x1, x2])[x1, x2]n−[x1, x2]nG([x1, x2])) = 0 (7)

for allx1, x2 ∈I.

Now we divide the proof in the two cases:

Case I : Let DandGare both Q-inner derivations ofR i.e.,D(x) = [b, x]for allx R and G(x) = [c, x]for allx∈R, whereb, c∈Q. Then from (7), we obtain that

a(b[x1, x2]n+1[x1, x2]b[x1, x2]n−[x1, x2]nc[x1, x2] + [x1, x2]n+1c) = 0 (8)

for allx1, x2 ∈I. By Chuang , this GPI is also satisfied byQand hence byR. By Lemma 2.4, if

Rdoes not satisfys4,b, c∈C, that is,D=G= 0, which is our conclusion (1). If char(R)6= 2and

Rsatisfiess4, then by Lemma 2.4,b+c ∈Cwhennis odd andb−c∈ Cwhennis even. Ifnis

even, thenb−c∈CimpliesD=G. Thus conclusions (2) and (3) are obtained.

Case II : Next assume that D andG are not bothQ-inner derivations of R, but they are C

-dependent modulo inner derivations ofR. SupposeD=λG+adb, that is,D(x) =λG(x) + [b, x]

for allx∈R, whereλ∈C,b∈Q. ThenGcannot be an inner derivation ofR. From (7), we obtain

thatI satisfies

a µ

λG([x1, x2])[x1, x2]n+ [b,[x1, x2]][x1, x2]n−[x1, x2]nG([x1, x2])

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SinceGis not inner derivation ofR, by Kharchenko’s theorem , we have that

a µ

λ([u, x2] + [x1, v])[x1, x2]n+ [b,[x1, x2]][x1, x2]n−[x1, x2]n([u, x2] + [x1, v])

= 0. (9)

for allx1, x2, u, v ∈I. By Chuang , this GPI is also satisfied byQand hence byR. In particular

foru=v= 0, we have thatRsatisfies

a[b,[x1, x2]][x1, x2]n= 0.

By Lemma 2.4, we get thatb∈C. Hence, we get from (9) thatRsatisfies

a µ

λ([u, x2] + [x1, v])[x1, x2]n−[x1, x2]n([u, x2] + [x1, v])

= 0. (10)

Assumingv = 0andu=x1, we have thatRsatisfies

a(λ−1)[x1, x2]n+1= 0 (11)

that is

a0[x1, x2]n+1 = 0, (12)

wherea0 = a(λ−1). For any two fixedx, y R, setw = [x, y]n+1. Thena0w = 0. From (12),

we can writea0[u, wva0]n+1 = 0for allu, v∈ R. Sincea0w = 0, it reduces toa0(uwva0)n+1 = 0.

This can be written as(wva0u)n+2 = 0for allu, v R. By Levitzki’s Lemma [15, Lemma 1.1],

wva0u= 0for allu, vR. SinceRis prime, eithera0 = 0orw= 0. Ifw= 0, then[x, y]n+1 = 0

for allx, y R. In this case by Herstein [13, Theorem 2],Ris commutative, contradicting the fact

that0 6= L is noncentral. Thusa0 = a(λ−1) = 0. Sincea 6= 0, this impliesλ = 1and hence D=G. Now (10) gives

a µ

([u, x2] + [x1, v])[x1, x2]n−[x1, x2]n([u, x2] + [x1, v])

= 0 (13)

for allu, v, x1, x2 R. SinceRis noncommutative, we may choosep ∈R such thatp /∈C. Then

replacinguwith[p, x1]andvwith[p, x2]in (13), we get

a µ

[p,[x1, x2]][x1, x2]n−[x1, x2]n[p,[x1, x2]]

= 0 (14)

for allx1, x2 ∈R. Then by Lemma 2.4, we conclude that sincep /∈C, char(R)6= 2,Rsatisfiess4

andnis even. Thus conclusion (2) is obtained.

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Next assume that DandGareC-independent modulo inner derivations ofR. Since neitherD

norGis inner, by Kharchenko’s theorem , we have from (7) thatIsatisfies

a(([u, x2] + [x1, v])[x1, x2]n−[x1, x2]n([x, x2] + [x1, y])) = 0. (15)

In particular,I satisfies the blended component

a[u, x2][x1, x2]n= 0. (16)

By , this GPI is also satisfied byQand hence byR. In particular, assumingu =x1, we getR

satisfiesa[x1, x2]n+1 = 0. SinceRis noncommutative, as above this leadsa= 0, a contradiction.2

Theorem 2.6 — Letnbe a positive integer,Rbe a noncommutative prime ring with char(R)6= 2, DandGtwo derivations ofR. Suppose that there exists06=a∈Rsuch thata(D(x)xn−xnG(x)) = 0for allx∈R, wheren≥1is a fixed integer. ThenD=G= 0.

PROOF: By Theorem 2.5, we have only to consider the case whenRsatisfiess4. IfD=G= 0,

we are done. So, let one of them must be nonzero. By Theorem 2.5, we have the following two cases:

Case I :nis even andD=G.

LetDbeQ-inner, that isD(x) = [b, x]for allx∈Randb∈Q. ThenRsatisfies

a(bxn+1−xbxn−xnbx+xn+1b) = 0. (17)

SinceRsatisfiess4, there exists a fieldKsuch thatR⊆M2(K)andM2(K)satisfies (l7). Since

M2(F)is a dense ring ofK-linear transformations over a vector spaceV, we assume that there exists v6= 0, such that{v, bv}is linearK-independent. By the density ofM2(K), there existr ∈M2(K)

such thatrv= 0, rbv=bv. Then

0 =a(brn+1−rbrn−rnbr+rn+1b)v=abv.

Of course for any u V, {u, v}linearly K-dependent impliesabu = 0. Letab 6= 0. Then

there existsw∈V such thatabw 6= 0and so{w, v}are linearlyK-independent. Alsoab(w+v) = abw6= 0andab(w−v) =abw6= 0. By the above argument, it follows thatwandbware linearlyK

-dependent, as are{w+v, b(w+v)}and{w−v, b(w−v)}. Therefore there existαw, αw+v, αw−v ∈K

such that

bw=αww, b(w+v) =αw+v(w+v), b(w−v) =αw−v(w−v).

In other words we have

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and

αww−bv=αwvw−αwvv. (19)

By comparing (18) with (19) we get both

(2αw−αw+v−αw−v)w+ (αw−v−αw+v)v= 0 (20)

and

2bv= (αw+v−αw−v)w+ (αw+v+αw−v)v. (21)

By (20), and since{w, v}areK-independent andchar(K) 6= 2, we haveαw =αw+v =αw−v.

Thus by (21) it follows2bv = 2αwv. This leads a contradiction with the fact that{v, bv}is linear

K-independent.

Therefore, we conclude thatab = 0. Again, by the density ofM2(K), there existr M2(K)

such thatrv= 0, rbv=v+bv. Then

0 =a(brn+1−rbrn−rnbr+rn+1b)v =av. By above argument, this impliesa= 0, a contradiction.

LetDbe notQ-inner, then by Kharchenko’s Theorem ,Rsatisfies

a(yxn−xny) = 0. (22)

ThusR ⊆M2(K)andM2(K)satisfies (22). Assumingy =e12, x= e11, we get0 = −ae12,

implyinga11=a21 = 0. Again, assumingy =e21, x=e11, we havea22 =a12= 0, that isa= 0,

Case II :nis odd andD(x) = [b, x], G(x) = [c, x]for allx∈Rwithb+c∈C.

In this case by hypothesis,Rsatisfies

a(bxn+1−xbxn+xnbx−xn+1b) = 0. (23)

By the above argument of case-I, this impliesa= 0, a contradiction. 2

3. THERESULTS ONSEMIPRIMERINGS ANDBANACHALGEBRAS

In this section we extended Theorem 2.6 to the semiprime ring. LetR be a semiprime ring andU

be its left Utumi quotient ring. ThenC =Z(U)is the extended centroid ofR([8, p-38]). It is well

known that any derivation of a semiprime ringRcan be uniquely extended to a derivation of its left

Utumi quotient ringU and so any derivation ofRcan be defined on the whole ofU [20, Lemma 2].

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By the standard theory of orthogonal completions for semiprime rings, we have the following

lemma.

Lemma 3.1 — (, Lemma 1 and Theorem 1). LetRbe a 2-torsion free semiprime ring andP a

maximal ideal ofC. ThenP U is a prime ideal ofU invariant under all derivations ofU. Moreover,

T

{P U :P ∈M(C) with U/P U 2−torsion free}= 0.

Theorem 3.2 — Let R be a 2-torsion free noncommutative semiprime ring, U the left Utumi

quotient ring ofR,da nonzero derivation ofRand0 6=a∈ R. Ifad(R) ⊆Z(R), then there exist

orthogonal central idempotentse1,e2,e3 ∈U withe1+e2+e3= 1such thatd(e1U) = 0,e2a= 0,

ande3U is commutative.

PROOF: Since any derivation dcan be uniquely extended to a derivation in U, andU andR

satisfy the same differential identities ( see ), thenad(x)∈Z(U), for allx∈U. LetP ∈M(C)

such thatU/P U is 2-torsion free. Note thatU is also 2-torsion free semiprime ring. P U is a prime

ideal ofU, which is d−invariant by Lemma 3.1 denote U = U/P U anddthe derivation induced

bydon U. For anyx U,ad(x) Z(U). SinceZ(U) = (C+P U)/P U = C/P U,ad(x) (C+P U)/P U = C/P U, for allx U. MoreoverU is a prime ring. Then it is clear that either d = 0; ora = 0inU; or[U , U] = 0. This implies that, for anyP M(C), eitherd(U) P U;

ora∈P U; or[U, U]⊆P U. In any casead(U)[U, U] P U for anyP M(C). By Lemma 3.1,

T

{P U :P ∈M(C) with U/P U 2torsion free}= 0. Thusad(U)[U, U] = 0.

By using the theory of orthogonal completion for semiprime rings ( see, [3, Chapter 3]), it follows

that there exist orthogonal central idempotents e1, e2, e3 U with e1 +e2 +e3 = 1 such that

d(e1U) = 0,e2a= 0, ande3U is commutative. 2

Theorem 3.3 — Letnbe a positive integer,Rbe noncommutative2-torsion free semiprime ring, U the left Utumi quotient ring ofRand0 6= a∈ R. Letd, gbe two nonzero derivations ofRsuch

thata(d(x)xn−xng(x)) = 0for allx∈R. Then there exist orthogonal central idempotentse1,e2,

e3 ∈U withe1+e2+e3 = 1such thatd(e1U) =g(e1U),e2a= 0, ande3U is commutative.

PROOF: Since any derivation dcan be uniquely extended to a derivation in U, andU andR

satisfy the same differential identities (see ), thena(d(x)xn−xng(x)) = 0, for allx ∈U. Let P ∈M(C)such thatU/P U is 2-torsion free. Note thatU is also 2-torsion free semiprime ring. By

Lemma 3.1P Uis a prime ideal ofU, which isd−invariant. DenoteU =U/P U anddthe derivation

induced bydonU. For anyx U, we geta(d(x)xn−xng(x)) = 0. MoreoverU is a prime ring

so by Theorem 2.6 we get eitherd = 0andg = 0 or[U , U] = 0, ora = 0. In any case we both

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P M(C) with U/P U 2−torsion free} = 0. Thenad(U)[U, U] = 0 andag(U)[U, U] = 0.

obtain thatad(R) ⊆Z(R)andag(R) ⊆Z(R). Hence, we geta(d−g)(R)⊆Z(R). By Theorem

3.2, we have the conclusion. 2

Corollary 3.4 — LetR be a noncommutative 2-torsion free semiprime ring, U the left Utumi

quotient ring ofRand06=a∈R. Letdbe a nonzero derivation ofRsuch thata(d(x)xn+xnd(x)) = 0for allx∈R. Then there exist orthogonal central idempotentse1,e2,e3 ∈U withe1+e2+e3= 1

such thatd(e1U) = 0,e2a= 0, ande3U is commutative.

By a Banach algebra, we shall mean a complex normed algebraAwhose underlying vector space

is Banach space. The Jacobson radical ofA is the intersection of all primitive ideals ofA and is

Theorem 3.5 — LetAbe a Banach algebra,d, ga pair of continuous Jordan derivations ofA,

PROOF : Let P be any primitive ideal of A. Sinced, g are both continuous, d(P) P and g(P) P by Lemma 3.2 in . Then dandgcan be induced to a pair of Jordan derivations on

Banach algebraA/P as follows :

d(x) =d(x) +P and g(x) =g(x) +P

for all x A/P and x A. Since P is a primitive ideal ofA, quotient algebra A/P is prime

and semisimple. On the other hand, we should remark that the pair of Jordan derivationsdandg

onA/P are also a pair of derivations onA/P by Breˇsar . Moreover, by a result of Johnson and

Sinclair  every derivation on a semisimple Banach algebra is continuous. Hence there are no

nonzero continuous derivations on commutative semisimple Banach algebras. Hence, we getd= 0

andg= 0whenA/P is commutative. It remains to show thatd= 0andg= 0in case of whenA/P

is noncommutative. By the hypothesis we have

a(d(x)xnxng(x)) = 0

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Corollary 3.6 — LetAa semisimple Banach algebra, d, gbe a pair of Jordan derivations onA

and06=a∈ A. Ifa(d(x)xn−xng(x)) = 0for allx ∈A, wherenis a fixed positive integer, then d= 0andg= 0.

REFERENCES

1. N. Argac¸ and V. De Filippis, Co-centralizing derivations and nilpotent values on Lie ideals, Indian J.

Pure Appl. Math., 41(3) (2010), 475-483.

2. K. I. Beidar, Rings of quotients of semiprime rings, Vestnik Moskov. Univ. Ser. I Mat. Mekh. (Engl.

Transl: Moscow Univ. Math. Bull.), 33 (1978), 36-42.

3. K. I. Beidar, W. S. Martindale III and A. V. Mikhalev, Rings with generalized identities, Pure and Applied

Math., 196, Marcel Dekker, New York, 1996.

4. J. Bergen, I. N. Herstein and J. W. Kerr, Lie ideals and derivations of prime rings, J. Algebra, 71 (1981),

259-267.

5. M. Breˇsar, Centralizing mappings and derivations in prime rings, J. Algebra, 156 (1993), 385-394.

6. L. Carini and V. De Filippis, Identities with generalized derivations on prime rings and Banach algebras,

Algebra Colloquium, 19(spec. 1) (2012), 971-986.

7. I. S. Chang, K. W. Jun and Y. S. Jung, On derivations in Banach algebras, Bull. Korean Math. Soc., 39

(4) (2002), 635-643.

8. C. L. Chuang, Hypercentral derivations, J. Algebra, 166(1) (1994), 34-71.

9. C. L. Chuang, GPIs having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc., 103(3) (1988),

723-728.

10. B. Dhara, Annihilator condition on power values of derivations, Indian J. Pure Appl. Math., 42(5)

(2011), 357-369.

11. B. Dhara, Power values of derivations with annihilator conditions on Lie ideals in prime rings, Comm.

Algebra, 37(6) (2009), 2159-2167.

12. T. S. Erickson, W. S. Martindale III and J. M. Osborn, Prime nonassociative algebras, Pacific J. Math.,

60 (1975), 49-63.

13. I. N. Herstein, Center-like elements in prime rings, J. Algebra, 60 (1979), 567-574.

14. I. N. Herstein, Rings with involution, Univ. of Chicago Press, Chicago, 1976.

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16. B. E. Johnson and A. M. Sinclair, Continuity of derivations and a problem of Kaplansky, Amer. J. Math.,

90 (1968), 1067-1073.

17. V. K. Kharchenko, Differential identity of prime rings, Algebra and Logic., 17 (1978), 155-168.

18. C. Lanski and S. Montgomery, Lie structure of prime rings of characteristic2, Pacific J. Math., 42 (1)

(1972), 117-136.

19. T. K. Lee and Y. Zhou, An identity with generalized derivations, Journal of Algebra and its Applications,

8(3) (2009), 307-317.

20. T. K. Lee, Semiprime rings with differential identities, Bull. Inst. Math. Acad. Sinica, 20(1) (1992),

27-38.

21. P. H. Lee and T. L. Wong, Derivations cocentralizing Lie ideals, Bull. Inst. Math. Acad. Sinica, 23

(1995), 1-5.

22. W. S. Martindale III, Prime rings satisfying a generalized polynomial identity, J. Algebra, 12 (1969),

576-584.

23. E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc., 8 (1957), 1093-1100.

24. A. M. Sinclair, Jordan homomorphisms and derivations on semisimple Banch algebras, Proc. Amer.

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