yk2Z.R/for allx

Vol. 19 (2018), No. 2, pp. 899–905 DOI: 10.18514/MMN.2018.1529

AN IDENTITY WITH DERIVATIONS IN PRIME RINGS

SHULIANG HUANG Received 03 February, 2015

Abstract. LetR be a prime ring with center Z.R/, andd a derivation ofR. Suppose that .d Œx; y_k/ⁿ mŒx; y_k2Z.R/for allx; y2R, wherem¤n; k1are fixed integers. ThendD 0orRsatisfiess4, the standard identity in four variables. In the case.d Œx; y_k/ⁿ mŒx; y_kD0 for allx; y2R, thendD0orRis commutative.

2010Mathematics Subject Classification: 16N60; 16W25; 16U80; 16D90 Keywords: prime and semiprime rings, derivations, GPIs

1. INTRODUCTION

In all that follows, unless stated otherwise,Rwill be an associative ring,Z.R/the center ofR,Q its Martindale quotient ring and Utumi quotient ringU. The center ofQ, denoted byC, is called the extended centroid ofR(we refer the reader to [3]

for related symbols). For any x; y 2R, the symbol Œx; yand xıy stand for the commutatorxy yxand anti-commutatorxyCyx, respectively. Recall that a ring Ris prime if for anya; b2R,aRbD.0/impliesaD0orbD0and is semiprime if for anya2R,aRaD f0gimpliesaD0. By a derivation onRwe mean an additive mappingdWR !Rsuch thatd.xy/Dd.x/yCxd.y/holds for allx; y2R. In par- ticulard is called an inner derivation induced by an elementa2R, ifd.x/DŒa; x

for all x2R: For any x; y 2R, we setŒx; y1DŒx; yDxy yx, andŒx; yk D ŒŒx; yk 1; y, wherek > 1is an integer. Note thatŒx; ykDPk

iD0. x/ⁱyx^{k i} and d.Œx; yk/DŒd.x/; ykCPk

iD1ŒŒŒx; yi 1; d.y/; yk i.

Many results in literature indicate that global structure of a ringRis often lightly connected to the behavior of additive mappings defined onR. The first classic result on this topic is due to Divinsky [7] who proved that a simple Artinian ring is commut- ative if it has a commuting non-identity automorphism. Over the last few decades, a number of authors have investigated the relationship between the commutativity of

The author was supported by the Anhui Provincial Natural Science Foundation (Grant No.

1808085MA14) and the Key University Science Research Project of Anhui Province (Grant No.

KJ2018A0433) and Research Project of Chuzhou University (Grant No. zrjz2017005) of P.R. China.

c 2018 Miskolc University Press

the ringRand certain specific types of derivations ofR. In [2], Ashraf and Rehman proved that ifRis a prime ring, I a nonzero ideal ofR andd is a derivation ofR such thatd.xıy/Dxıyfor allx; y2I, thenRis commutative. In [1], Argac¸ and Inceboz generalized the above result as following: LetRbe a prime ring,Ia nonzero ideal ofRandna fixed positive integer, ifRadmits a derivationd with the property .d.xıy//ⁿDxıyfor allx; y2I, thenRis commutative. On the other hand, Daif and Bell [6] showed that if in a semiprime ringR there exists a nonzero idealI of Rand a derivationd such thatd.Œx; y/DŒx; yfor allx; y2I, thenI Z.R/. In particular, ifIDRthenRis commutative. Motivated by the above-cited results, our purpose in this article is to obtain some information of the prime ringR involving a central identity.d Œx; y_k/ⁿ mŒx; y_k 2Z.R/for allx; y2R, wherem; n; k1 are fixed integers.

The standard identitys4in four variables is defined as follows:

s4DX

. 1/X _.1/X _.2/X _.3/X _.4/

where. 1/ is the sign of a permutation of the symmetric group of degree 4. As is well known, prime rings satisfyings4can be characterized by the following:

Fact 1([4]). LetR be a prime ring with the extended centroidC. Then the fol- lowing are equivalent:

(1)d i mCRC4;

(2)Rsatisfiess4;

(3)Ris commutative orRembeds inM2.F /;

(4)Ris algebraic of bounded degree 2 overC; (5)RsatisfiesŒŒx²; y; Œx; y.

2. RESULTS

We begin with the following lemmas which are crucial for proving our main res- ults.

Lemma 1. LetRDM2.F / be the ring of all22matrices over a field F. If 0¤a2Rsuch that

.ŒŒa; x; y_kC

k

X

iD1

ŒŒŒx; yi 1; Œa; y; y_{k i}/ⁿDmŒx; y_k

for allx2Rwherem; n; k1are fixed integers, thena2Z.R/.

Proof. Let aDP2

i;jD0aijeij with aij 2F, whereeij is the usual matrix unit with 1 in.i; j /-entry and zero elsewhere. LetxDe12,yDe11. ThenmŒx; y_k D 0and.ŒŒa; x; y_kCPk

iD1ŒŒŒx; yi 1; Œa; y; y_{k i}/ⁿD. 1/^{k n}. e12aCe11ae12/ⁿ: By assumption we have. 1/^{k n}. e12aCe11ae12/ⁿD0:Right multiplying bye12, it yields that . 1/^.kC1/na₂₁ⁿ e12 D0, which implies a21D0. Similarly, we have a12D0. Thusamust be a diagonal matrix. Now setaDP

tat tet t, withat t2F. Let' be the inner automorphism ofR given by '.x/D.1Ceij/a.1 eij/. Thus .ŒŒa^'; x; y_kCPk

iD1ŒŒŒx; yi 1; Œa^'; y; y_{k i}/ⁿDmŒx; y_k for all x; y 2R. By above argument,

a^'D.1Ceij/a.1 eij/D˙_i^k_D1ai iei iC.ajj ai i/eij

must be diagonal. Thereforeajj Dai i and soa2Z.R/.

Lemma 2. LetRbe a non-commutative prime ring with centerZ.R/. If0¤a2R such that

.ŒŒa; x; ykC

k

X

iD1

ŒŒŒx; yi 1; Œa; y; yk i/ⁿDmŒx; yk

for allx; y2R, wheren; m; k1are fixed integers, thena2Z.R/.

Proof. By assumption,Rsatisfies the generalized polynomial identity p.x; y/D.ŒŒa; x; ykC

k

X

iD1

ŒŒŒx; yi 1; Œa; y; yk i/ⁿ mŒx; yk:

By Chuang [5], this generalized polynomial identity (GPI) is also satisfied byU. Ifa…C thenp.x; y/D0is a nontrivial (GPI) forU. In caseC is infinite, we have p.x; y/D0 for all x; y2UN

CC whereC is the algebraic closure of C. Since bothU andUN

CC are prime and centrally closed [8], we may replaceRbyU or UN

CC according toC finite or infinite. Thus we may assume thatR is centrally closed overC which is either finite or algebraically closed andp.x; y/D0for all x; y2R. By Martindale’s theorem [13],R is then a primitive ring having nonzero soc.R/with C as the associated division ring. Hence by Jacobson’s theorem [9], R is isomorphic to a dense ring of linear transformations of a vector spaceV over C. If dimCV Dk, then the density of R on V implies that RŠM_k.C /, where kDdimCV. SinceRis noncommutative,k2. If dimCV D2, then by Lemma1we

havea2Z.R/. Now suppose that dimCV 3. We want to show that for anyv2V, v andav are linearlyC-dependent. Suppose on contrary thatv andav are linearly C-independent for somev 2V. Since dimCV 3, there exists w2V such that fv; av; wgare linearlyC-independent set of vectors. By the density ofRonV, there existx; y2Rsuch thatxvDv; xwDvCw; xavDwIyvDv; ywD0; yavDv:

Then 0Dp.x; y/vD. 1/^{k n}v; a contradiction. Therefore v, av are linearly C- dependent for anyv2V. Hence we can writeavDv˛v for allv2V and˛v2C. Then by a standard argument, it is very easy to prove that˛v is independent of the choice of v2 V. In fact, Since d i mCV 3, then there exist u; v; w which are linearly independent, and so˛u; ˛v; ˛w2C such thatauDu˛u,avDv˛v,awD w˛w, that isa.uCvCw/Du˛uCv˛vCw˛w. Moreovera.uCvCw/D.uC vCw/˛uCvCw for some˛uCvCw 2C. Then0D.˛uCvCw ˛u/uC.˛uCvCw

˛_v/vC.˛_uCvCw ˛_w/w and hence˛_uD˛_vD˛_wD˛_uCvCw, that is˛ does not depend on the choice ofv. Thus we can write avDv˛ for all v2V and˛ 2C fixed. Now, letr2R, v2V. Since avDv˛, we haveŒa; rvD.ar/v .ra/vD a.rv/ r.av/D.rv/˛ r.v˛/D0;that is Œa; rV D0. Hence Œa; rD0 for all

r2R, implyinga2Z.R/.

Theorem 1. Let R be a prime ring and d a derivation of R. Suppose that .d Œx; y_k/ⁿDmŒx; y_kfor allx; y2R, wherem¤n; k1are fixed integers. Then d D0orRis commutative.

Proof. Using the identityd.Œx; yk/DŒd.x/; ykCPk

iD1ŒŒŒx; yi 1; d.y/; yk i

and the hypothesis, we have .Œd.x/; y_kC

k

X

iD1

ŒŒŒx; yi 1; d.y/; y_{k i}/ⁿDmŒx; y_k

for allx; y2R. Assume first thatd isQ-inner, that is,d.x/DŒa; xfor allx2Q, whereais a non-central element inQ. Then

.ŒŒa; x; y_kC

k

X

iD1

ŒŒŒx; yi 1; Œa; y; y_{k i}/ⁿDmŒx; y_k for allx; y2U. Thus by Lemma2,a2Z.R/that givesd D0.

Assume next thatd isQ-outer. Applying Kharchenko’s theorem [10], we get .Œv; y_kCPk

iD1ŒŒŒx; yi 1; u; y_{k i}/ⁿDmŒx; y_kfor allx; y; u; v2U. In partic- ular, foruD0andvDxwe have.Œx; y_k/ⁿDmŒx; y_k;for allx; y2U. Note that, this is a polynomial identity and hence there exists a fieldF such thatRMt.F /, the ring ofttmatrices over a fieldF, wheret1. By Chuang [5], this generalized polynomial identity (GPI) is also satisfied byRas well. Moreover,RandMt.F /sat- isfy the same polynomial identity [11, Lemma 1], that is,.Œx; y_k/ⁿDmŒx; y_kfor all x; y2M_t.F /. But by choosingxDe₁₂,yDe₁₁, we get0D.Œx; y_k/ⁿ mŒx; y_kD . 1/^k.e₁₂ⁿ me12/:This is a contradiction, ending the proof.

Lemma 3. LetRDMt.F /be the ring of alltt matrices over a fieldF with t 3. If 0¤a2 R and m; n; k 1 are fixed integers, such that .ŒŒa; x; y_kC Pk

iD1ŒŒŒx; yi 1; Œa; y; yk i/ⁿ mŒx; yk2FIt;for allx; y2R, thena2FIt. Proof. We are given that

Œ.ŒŒa; x; y_kC

k

X

iD1

ŒŒŒx; yi 1; Œa; y; y_{k i}/ⁿ mŒx; y_k; ´D0

for allx; y; ´2R. LetaD.aij/tt. By choosingxDeij; yDei i and´Dei k for anyi¤j ¤k, we have

Œ. 1/^{k n}Œa; eijⁿ . 1/^kmeij; e_{i k}D. 1/^{k n}..eija/ⁿe_{i k} e_{i k}.aeij/ⁿ/D0:

Thusaij D0, and soais a diagonal matrix. Using the same technique in Lemma1,

we geta2FIt, proving the lemma.

Theorem 2. LetR be a prime ring with centerZ.R/, andd a derivation ofR.

Suppose that.d Œx; y_k/ⁿ mŒx; y_k2Z.R/for allx; y2R, wherem¤n; k1are fixed integers. ThendD0orRsatisfiess4, the standard identity in four variables.

Proof. By assumptionRsatisfies the generalized differential identity

0DŒ.d Œx; y_k/ⁿ mŒx; y_k; wDŒŒd.x/; y_kC

k

X

iD1

ŒŒŒx; yi 1; d.y/; y_{k i} mŒx; y_k; w

forx; y; w2R. By Lee [12],RandU satisfy the same differential identities we may assume that above identity is also satisfied byU. Now we consider the following two cases:

Case 1. Suppose thatd is aQ-outer derivation. Then Kharchenko’s theorem [10], we have

Œ.Œv; ykC

k

X

iD1

ŒŒŒx; yi 1; u; yk i/ⁿ mŒx; yk; wD0;

for allx; y; u; v; w2U. This is a polynomial identity and hence there exists a field F such that U Mt.F / with t > 1 and U, Mt.F /satisfy the same polynomial identity [11]. Ift 3then by choosingwDe13,vDxDe12,yDe11,uD0, we get0DŒ.Œv; y_kCPk

iD1ŒŒŒx; yi 1; u; y_{k i}/ⁿ mŒx; y_k; wD. 1/^ke13:This is a contradiction. ThustD2and soRsatisfiess4by Fact1.

Case 2. Suppose thatd is a Q-inner derivation. In this case there existsa2Q such thatd.x/DŒa; xfor allx2R. Then we have

Œ.ŒŒa; x; y_kC

k

X

iD1

ŒŒŒx; y_{i 1}; Œa; y; y_{k i}/ⁿ mŒx; y_k; wD0;

for allx; y; w2R. By localizingRatZ.R/it follows that .ŒŒa; x; y_kC

k

X

iD1

ŒŒŒx; yi 1; Œa; y; y_{k i}/ⁿ mŒx; y_k 2Z.RZ/;

for all x; y 2R_Z. SinceR andR_Z satisfy the same polynomial identities, in or- der to prove that R satisfies s4, we may assume that R is simple with 1. Hence, .ŒŒa; x; y_kCPk

iD1ŒŒŒx; yi 1; Œa; y; y_{k i}/ⁿ mŒx; y_k 2Z.R/for all x; y2R.

ThereforeRsatisfies a generalized polynomial identity and it is simple with1, which implies thatQDRC DR andRhas a minimal right ideal, whose commuting ring D is a division ring which is finite dimensional overZ.R/. However, sinceR is a simple ring with1,R must be Artinian, that is,RDD_s, thessmatrices overD, for somes1. By [11] there exists a fieldF such thatRMt.F /, the ring oftt matrices over fieldF, witht > 1, and

.ŒŒa; x; ykC

k

X

iD1

ŒŒŒx; yi 1; Œa; y; yk i/ⁿ mŒx; yk2Z.Mt.F //DFIt; for allx; y2Mt.F /. Ift3, then by Lemma3,a2FIt and sod D0. IftD2,

thenRsatisfiess4.

ACKNOWLEDGEMENT

The author would like to thank the referees for their helpful comments.

REFERENCES

[1] N. Argac¸ and H. G. Inceboz, “Derivations of prime and semiprime rings,”SIAM J. Korean Math.

Soc., vol. 46, no. 5, pp. 997–1005, 2009, doi:10.4134/JKMS.2009.46.5.997.

[2] M. Ashraf and N. Rehman, “On commutativity of rings with derivations,”SIAM Results Math., vol. 42, no. 1-2, pp. 3–8, 2002, doi:https://doi.org/10.1007/BF03323547.

[3] K. I. Beidar, W. S. Martindale III, and A. V. Mikhalev,Rings with generalized identities. New York: Marcel Dekker, 1996. doi:10.1007/978-1-4614-6946-9.

[4] M. Bresar, “Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings,” SIAM Tran. Amer. Math. Soc., vol. 335, no. 2, pp. 525–546, 1993, doi:

10.2307/2154392.

[5] C. L. Chuang, “GPIs having coefficients in Utumi quotient rings,”SIAM Proc. Amer. Math. Soc., vol. 103, no. 3, pp. 723–728, 1988, doi:10.2307/2046841.

[6] M. N. Daif and H. E. Bell, “GPIs Remarks on derivations on semiprime rings,”SIAM Int. J. Math.

Math. Sci., vol. 15, no. 1, pp. 205–206, 1992, doi:10.1155/S0161171292000255.

[7] N. Divinsky, “On commuting automorphisms of rings,”SIAM Proc. Trans. R. Soc. Canada III (3), vol. 49, no. 3, pp. 19–22, 1955.

[8] T. S. Erickson, W. S. Martindale, and J. M. Osborn, “Prime nonassociative algebras,”SIAM Pacific J. Math., vol. 60, no. 1, pp. 49–63, 1975, doi:10.2140/pjm.1975.60.49.

[9] N. Jacobson,Structure of rings. New York: Marcel Dekker, 1964. doi:10.1090/s0002-9904- 1957-10071-8.

[10] V. K. Kharchenko, “Differential identity of prime rings,”SIAM Algebra and Logic, vol. 17, no. 2, pp. 155–168, 1978, doi:https://doi.org/10.1007/BF01670115.

[11] C. Lanski, “An engle condition with derivation,”SIAM Proc. Amer. Math. Soc., vol. 183, no. 3, pp. 731–734, 1993, doi:10.2307/2160113.

[12] T. K. Lee, “Semiprime rings with differential identities,”SIAM Bull. Inst. Math. Acad. Sinica., vol. 20, no. 1, pp. 27–38, 1992.

[13] W. S. Martindale 3rd, “Prime rings satistying a generalized polynomial identity,”SIAM J. Algebra, vol. 12, no. 4, pp. 576–584, 1969, doi:https://doi.org/10.1016/0021-8693(69)90029-5.

Author’s address

Shuliang Huang

Chuzhou University, School of Mathematics and Finance, 239000 Chuzhou P. R. China E-mail address:shulianghuang@163.com