Shuliang, Huang

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Miskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 19 (2018), No. 2, pp. 899–905 DOI: 10.18514/MMN.2018.1529AN IDENTITY WITH DERIVATIONS IN PRIME RINGSSHULIANG HUANGReceived 03 February, 2015Abstract. Let R be a prime ring with center Z.R/, and d a derivation of R. Suppose that.d Œx;yk/n mŒx;yk 2Z.R/ for all x;y 2R, wherem¤ n;k  1 are fixed integers. Then d D0 or R satisfies s4, the standard identity in four variables. In the case .d Œx;yk/n mŒx;yk D 0for all x;y 2R, then d D 0 or R is commutative.2010 Mathematics Subject Classification: 16N60; 16W25; 16U80; 16D90Keywords: prime and semiprime rings, derivations, GPIs1. INTRODUCTIONIn all that follows, unless stated otherwise, R will be an associative ring, Z.R/ thecenter of R, Q its Martindale quotient ring and Utumi quotient ring U . The centerof Q, denoted by C , is called the extended centroid of R (we refer the reader to [3]for related symbols). For any x;y 2 R, the symbol Œx;y and x ı y stand for thecommutator xy yx and anti-commutator xyCyx, respectively. Recall that a ringR is prime if for any a;b 2R, aRb D .0/ implies aD 0 or b D 0 and is semiprime iffor any a 2 R, aRaD f0g implies aD 0. By a derivation on R we mean an additivemapping d WR !R such that d.xy/D d.x/yCxd.y/ holds for all x;y 2R. In par-ticular d is called an inner derivation induced by an element a 2 R, if d.x/D Œa;xfor all x 2 R: For any x;y 2 R, we set Œx;y1 D Œx;y D xy   yx, and Œx;yk DŒŒx;yk 1;y, where k > 1 is an integer. Note that Œx;yk DPkiD0. x/iyxk i andd.Œx;yk/D Œd.x/;ykCPkiD1ŒŒŒx;yi 1;d.y/;yk i .Many results in literature indicate that global structure of a ring R is often lightlyconnected to the behavior of additive mappings defined on R. The first classic resulton 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 ofThe 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 Press900 SHULIANG HUANGthe ring R and certain specific types of derivations of R. In [2], Ashraf and Rehmanproved that if R is a prime ring, I a nonzero ideal of R and d is a derivation of Rsuch that d.x ıy/D x ıy for all x;y 2 I , then R is commutative. In [1], Argac¸ andInceboz generalized the above result as following: LetR be a prime ring, I a nonzeroideal of R and n a fixed positive integer, if R admits a derivation d with the property.d.x ıy//n D x ıy for all x;y 2 I , then R is commutative. On the other hand, Daifand Bell [6] showed that if in a semiprime ring R there exists a nonzero ideal I ofR and a derivation d such that d.Œx;y/D Œx;y for all x;y 2 I , then I Z.R/. Inparticular, if I DR then R is commutative. Motivated by the above-cited results, ourpurpose in this article is to obtain some information of the prime ring R involvinga central identity .d Œx;yk/n mŒx;yk 2 Z.R/ for all x;y 2 R, where m;n;k  1are fixed integers.The standard identity s4 in four variables is defined as follows:s4 DX. 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 iswell known, prime rings satisfying s4 can be characterized by the following:Fact 1 ([4]). Let R be a prime ring with the extended centroid C . Then the fol-lowing are equivalent:(1) dimCRC  4;(2) R satisfies s4;(3) R is commutative or R embeds in M2.F /;(4) R is algebraic of bounded degree 2 over C ;(5) R satisfies ŒŒx2;y; Œx;y.2. RESULTSWe begin with the following lemmas which are crucial for proving our main res-ults.AN IDENTITY WITH DERIVATIONS IN PRIME RINGS 901Lemma 1. Let R DM2.F / be the ring of all 2 2 matrices over a field F . If0¤ a 2R such that.ŒŒa;x;ykCkXiD1ŒŒŒx;yi 1; Œa;y;yk i /n DmŒx;ykfor all x 2R where m;n;k  1 are fixed integers, then a 2Z.R/.Proof. Let a DP2i;jD0aij eij with aij 2 F , where eij is the usual matrix unitwith 1 in .i;j /-entry and zero elsewhere. Let x D e12, y D e11. Then mŒx;yk D0 and .ŒŒa;x;ykCPkiD1ŒŒŒx;yi 1; Œa;y;yk i /nD . 1/kn. e12aCe11ae12/n:By assumption we have . 1/kn. e12aC e11ae12/n D 0: Right multiplying by e12,it yields that . 1/.kC1/nan21e12 D 0, which implies a21 D 0. Similarly, we havea12 D 0. Thus a must be a diagonal matrix. Now set a DPt at tet t , with at t 2 F .Let ' be the inner automorphism of R given by '.x/ D .1C eij /a.1  eij /. Thus.ŒŒa' ;x;yk CPkiD1ŒŒŒx;yi 1; Œa' ;y;yk i /n D mŒx;yk for all x;y 2 R. Byabove argument,a' D .1C eij /a.1  eij /D˙kiD1ai iei i C .ajj  ai i /eijmust be diagonal. Therefore ajj D ai i and so a 2Z.R/. Lemma 2. LetR be a non-commutative prime ring with centerZ.R/. If 0¤ a 2Rsuch that.ŒŒa;x;ykCkXiD1ŒŒŒx;yi 1; Œa;y;yk i /n DmŒx;ykfor all x;y 2R, where n;m;k  1 are fixed integers, then a 2Z.R/.Proof. By assumption, R satisfies the generalized polynomial identityp.x;y/D .ŒŒa;x;ykCkXiD1ŒŒŒx;yi 1; Œa;y;yk i /n mŒx;yk :By Chuang [5], this generalized polynomial identity (GPI) is also satisfied by U .If a … C then p.x;y/D 0 is a nontrivial (GPI) for U . In case C is infinite, we havep.x;y/ D 0 for all x;y 2 UNC C where C is the algebraic closure of C . Sinceboth U and UNC C are prime and centrally closed [8], we may replace R by U orUNC C according to C finite or infinite. Thus we may assume that R is centrallyclosed over C which is either finite or algebraically closed and p.x;y/ D 0 for allx;y 2 R. By Martindale’s theorem [13], R is then a primitive ring having nonzerosoc.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 space V overC . If dimCV D k, then the density of R on V implies that R Š Mk.C /, wherekDdimCV . SinceR is noncommutative, k  2. If dimCV D 2, then by Lemma 1 we902 SHULIANG HUANGhave a 2Z.R/. Now suppose that dimCV  3. We want to show that for any v 2 V ,v and av are linearly C -dependent. Suppose on contrary that v and av are linearlyC -independent for some v 2 V . Since dimCV  3, there exists w 2 V such thatfv;av;wg are linearly C -independent set of vectors. By the density of R on V , thereexist x;y 2 R such that xv D v;xw D vCw;xav D wIyv D v;yw D 0;yav D v:Then 0 D p.x;y/v D . 1/knv; a contradiction. Therefore v, av are linearly C -dependent for any v 2 V . Hence we can write av D v˛v for all v 2 V and ˛v 2 C .Then by a standard argument, it is very easy to prove that ˛v is independent of thechoice of v 2 V . In fact, Since dimCV  3, then there exist u;v;w which arelinearly independent, and so ˛u;˛v;˛w 2 C such that auD u˛u, av D v˛v, aw Dw˛w , that is a.uC vCw/ D u˛uC v˛vCw˛w . Moreover a.uC vCw/ D .uCvCw/˛uCvCw for some ˛uCvCw 2 C . Then 0D .˛uCvCw  ˛u/uC .˛uCvCw  ˛v/vC .˛uCvCw  ˛w/w and hence ˛u D ˛v D ˛w D ˛uCvCw , that is ˛ does notdepend on the choice of v. Thus we can write av D v˛ for all v 2 V and ˛ 2 Cfixed. Now, let r 2 R, v 2 V . Since av D v˛, we have Œa;rv D .ar/v  .ra/v Da.rv/  r.av/ D .rv/˛   r.v˛/ D 0; that is Œa;rV D 0. Hence Œa;r D 0 for allr 2R, implying a 2Z.R/. Theorem 1. Let R be a prime ring and d a derivation of R. Suppose that.d Œx;yk/n DmŒx;yk for all x;y 2R, wherem¤ n;k  1 are fixed integers. Thend D 0 or R is commutative.Proof. Using the identity d.Œx;yk/D Œd.x/;ykCPkiD1ŒŒŒx;yi 1;d.y/;yk iand the hypothesis, we have.Œd.x/;ykCkXiD1ŒŒŒx;yi 1;d.y/;yk i /n DmŒx;ykfor all x;y 2 R. Assume first that d is Q-inner, that is, d.x/D Œa;x for all x 2Q,where a is a non-central element in Q. Then.ŒŒa;x;ykCkXiD1ŒŒŒx;yi 1; Œa;y;yk i /n DmŒx;ykfor all x;y 2 U . Thus by Lemma 2, a 2Z.R/ that gives d D 0.Assume next that d is Q-outer. Applying Kharchenko’s theorem [10], we get.Œv;ykCPkiD1ŒŒŒx;yi 1;u;yk i /nDmŒx;yk for all x;y;u;v 2U . In partic-ular, for uD 0 and v D x we have .Œx;yk/n DmŒx;yk; for all x;y 2 U . Note that,this is a polynomial identity and hence there exists a field F such that R Mt .F /,the ring of t  t matrices over a field F , where t  1. By Chuang [5], this generalizedpolynomial identity (GPI) is also satisfied byR as well. Moreover,R andMt .F / sat-isfy the same polynomial identity [11, Lemma 1], that is, .Œx;yk/nDmŒx;yk for allx;y 2Mt .F /. But by choosing xD e12, yD e11, we get 0D .Œx;yk/n mŒx;yk D. 1/k.en12 me12/: This is a contradiction, ending the proof.AN IDENTITY WITH DERIVATIONS IN PRIME RINGS 903Lemma 3. Let R DMt .F / be the ring of all t  t matrices over a field F witht  3. If 0 ¤ a 2 R and m;n;k  1 are fixed integers, such that .ŒŒa;x;yk CPkiD1ŒŒŒx;yi 1; Œa;y;yk i /n mŒx;yk 2 F It ; for all x;y 2R, then a 2 F It .Proof. We are given thatŒ.ŒŒa;x;ykCkXiD1ŒŒŒx;yi 1; Œa;y;yk i /n mŒx;yk;´D 0for all x;y;´ 2 R. Let a D .aij /tt . By choosing x D eij ;y D ei i and ´D eik forany i ¤ j ¤ k, we haveŒ. 1/knŒa;eij n  . 1/kmeij ; eikD . 1/kn..eija/neik   eik.aeij /n/D 0:Thus aij D 0, and so a is a diagonal matrix. Using the same technique in Lemma 1,we get a 2 F It , proving the lemma. Theorem 2. Let R be a prime ring with center Z.R/, and d a derivation of R.Suppose that .d Œx;yk/n mŒx;yk 2Z.R/ for all x;y 2R, wherem¤ n;k  1 arefixed integers. Then d D 0 or R satisfies s4, the standard identity in four variables.Proof. By assumption R satisfies the generalized differential identity0D Œ.d Œx;yk/n mŒx;yk;wD ŒŒd.x/;ykCkXiD1ŒŒŒx;yi 1;d.y/;yk i mŒx;yk;wfor x;y;w 2R. By Lee [12], R and U satisfy the same differential identities we mayassume that above identity is also satisfied by U . Now we consider the following twocases:Case 1. Suppose that d is aQ-outer derivation. Then Kharchenko’s theorem [10],we haveŒ.Œv;ykCkXiD1ŒŒŒx;yi 1;u;yk i /n mŒx;yk;wD 0;for all x;y;u;v;w 2 U . This is a polynomial identity and hence there exists a fieldF such that U  Mt .F / with t > 1 and U , Mt .F / satisfy the same polynomialidentity [11]. If t  3 then by choosing w D e13, v D x D e12, y D e11, uD 0, weget 0D Œ.Œv;ykCPkiD1ŒŒŒx;yi 1;u;yk i /n mŒx;yk;w D . 1/ke13: This isa contradiction. Thus t D 2 and so R satisfies s4 by Fact 1.904 SHULIANG HUANGCase 2. Suppose that d is a Q-inner derivation. In this case there exists a 2Qsuch that d.x/D Œa;x for all x 2R. Then we haveŒ.ŒŒa;x;ykCkXiD1ŒŒŒx;yi 1; Œa;y;yk i /n mŒx;yk;wD 0;for all x;y;w 2R. By localizing R at Z.R/ it follows that.ŒŒa;x;ykCkXiD1ŒŒŒx;yi 1; Œa;y;yk i /n mŒx;yk 2Z.RZ/;for all x;y 2 RZ . Since R and RZ 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k CPkiD1ŒŒŒx;yi 1; Œa;y;yk i /n mŒx;yk 2 Z.R/ for all x;y 2 R.Therefore R satisfies a generalized polynomial identity and it is simple with 1, whichimplies that Q D RC D R and R has a minimal right ideal, whose commuting ringD is a division ring which is finite dimensional over Z.R/. However, since R is asimple ring with 1, R must be Artinian, that is, R DDs , the s s matrices over D,for some s  1. By [11] there exists a field F such that R Mt .F /, the ring of t  tmatrices over field F , with t > 1, and.ŒŒa;x;ykCkXiD1ŒŒŒx;yi 1; Œa;y;yk i /n mŒx;yk 2Z.Mt .F //D F It ;for all x;y 2Mt .F /. If t  3, then by Lemma 3, a 2 F  It and so d D 0. If t D 2,then R satisfies s4. ACKNOWLEDGEMENTThe 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. NewYork: Marcel Dekker, 1996. doi: 10.1007/978-1-4614-6946-9.[4] M. Bresar, “Commuting traces of biadditive mappings, commutativity-preserving mappingsand 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.AN IDENTITY WITH DERIVATIONS IN PRIME RINGS 905[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 PacificJ. 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 addressShuliang HuangChuzhou University, School of Mathematics and Finance, 239000 Chuzhou P. R. ChinaE-mail address: shulianghuang@163.com

An identity with derivations in prime rings

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