1、Chin.Quart.J.of Math.2023,38(2):134144Differential Identities in Prime Rings with InvolutionHUANG Shu-liang(School of Mathematics and Finance,Chuzhou University,Chuzhou 239000,China)Abstract:LetRbe a prime ring of characteristic different from two with the secondinvolutionandan automorphism ofR.An a
2、dditive mappingFofRis called ageneralized(,)-derivation onRif there exists an(,)-derivationdofRsuch thatF(xy)=F(x)(y)+(x)d(y)holds for allx,yR.The paper deals with the study ofsome commutativity criteria for prime rings with involution.Precisely,we describe thestructure ofRadmitting a generalized(,)
3、-derivationFsatisfying any one of thefollowing properties:(i)F(xx)(xx)Z(R),(ii)F(xx)+(xx)Z(R),(iii)F(x)F(x)(xx)Z(R),(iv)F(x)F(x)+(xx)Z(R),(v)F(xx)F(x)F(x)Z(R),(vi)F(xx)F(x)F(x)=0for allxR.Also,some examples are given to demonstrate that the restriction of thevarious results is not superfluous.In fac
4、t,our results unify and extend several well knowntheorems in literature.Keywords:Prime rings;Generalized(,)-derivations;Involution;Commutativity2000 MR Subject Classification:16W25,16N60,16U80CLC number:O153.3Document code:AArticle ID:1002-0462(2023)02-0134-11DOI:10.13371/ki.chin.q.j.m.2023.02.0031.
5、IntroductionThroughout the present paper,Rwill represent an associative ring with centerZ(R).Foranyx,yR,the symbol x,y andxystand for Lie productxyyxand Jordan productxy+yx,respectively.An additive subgroupJofRis called a Jordan ideal ofRifjrfor alljJandrR.An additive subgroupUofRis called a Lie ide
6、al ofRif u,rUfor alluUandrR,a Lie idealUis called square-closed ifu2Ufor alluU.A ringRis called 2-torsionReceived date:2022-04-18Foundation item:Supported by the University Science Research Project of Anhui Province(Grant Nos.KJ2020A0711,KJ2020ZD74,KJ2021A1096)and the Natural Science Foundation of A
7、nhui Province(Grant No.1908085MA03).Biographies:HUANG Shu-liang(1981-),male,native of Weifang,Shandong,professor of Chuzhou University,engages in rings and algebras.Corresponding author HUANG Shu-liang:.134No.2HUANG Shu-liang:Differential Identities in Prime Rings with Involution135free,if whenever
8、2x=0,withxR,thenx=0.Recall that a ringRis prime if for anya,bR,aRb=(0)impliesa=0 orb=0 and is semiprime if for anyaR,aRa=0impliesa=0.It isstraightforward to check that a prime ring of characteristic different from two is 2-torsion free.By a derivation onRwe mean an additive mappingd:RRthat satisfies
9、 the Leibnizruled(xy)=d(x)y+xd(y)holds for allx,yR.An additive mappingF:RRis called ageneralized derivation if there exists a derivationd:RRsuch thatF(xy)=F(x)y+xd(y)holds for all x,yR,and d is called the associated derivation of F.Letandbe automorphisms ofR.Set x,y,=x(y)(y)xand(xoy),=x(y)+(y)x.Foll
10、owing 4,an additive mappingd:RRis said to be an(,)-derivation ifd(xy)=d(x)(y)+(x)d(y)holds for allx,yR.An additive mappingF:RRis calleda generalized(,)-derivation onRif there exists an(,)-derivationd:RRsuch thatF(xy)=F(x)(y)+(x)d(y)holds for allx,yR.SetF(x)=a(x)+(x)bfor allxR,wherea and b are fixed
11、elements in R,then,F(xy)=a(x)(y)+(x)(y)b=(a(x)+(x)b)(y)+(x)(y)bb(y).It is clear thatFis a generalized(,)-derivation ofR.One may observe that the notionof generalized(,)-derivation includes those of(,)-derivation whenF=d,of derivationwhenF=dand=IR,and of generalized derivation,which is the case when=
12、IR.Thus,it should be interesting to extend some results concerning these notions to generalized(,)-derivations.An involutionof a ringRis an anti-automorphism of order 2(i.e.anadditive mapping satisfying(xy)=yxand(x)=xfor allx,yR).An elementrin a ringwith involution(R,)is called to be Hermitian ifr=r
13、and skew-Hermitian ifr=r.Thesets of all Hermitian and skew-Hermitian elements ofRwill be denoted byH(R)andS(R),respectively.The involution is said to be the first kind ifZ(R)H(R),otherwise it is said tobe of the second kind.In the latter caseS(R)TZ(R)6=0.Following 5,ifRis 2-torsion freethen everyxRc
14、an be uniquely represented in the form of 2x=h+k,wherehH(R)andkS(R).Moreover,in this casexis normal,i.e.,x,x=0,if and only ifhandkcommute.If all elements inRis normal,thenRis said to be a normal ring.A classical example is thering of Hamilton quaternions.Especially,derivations,biderivations and supe
15、rbiderivations ofquaternion rings were characterized in 15.It is also worthwhile to note that every primering having an involutionis-prime(i.e.aRb=aRb=(0)yields thata=0 orb=0)but theconverse is in general not true.A typical example in 24 is as following:LetRbe a prime ring,S=RRwhereRis the opposite
16、ring ofR,define(x,y)=(y,x).From(0,x)S(x,0)=0,it follows thatSis not prime.For the-primeness ofS,we suppose that(a,b)S(x,y)=0 and(a,b)S(x,y)=0,then we get(aRx,yRb)=0 and(aRy,xRb)=0,and henceaRx=yRb=aRy=xRb=0,or equivalently(a,b)=0 or(x,y)=0.This example shows that every prime ring can be injected in
17、a-prime ring and from thispoint of view-prime rings constitute a more general class of prime rings.Moreover,some136CHINESE QUARTERLY JOURNAL OF MATHEMATICSVol.38well known results on prime and semiprime rings have been extended to rings with involution(see 7,16,20 and 22,where further references can
18、 be found).Many results in literature indicate that the global structure of a ringRis often lightlyconnected to the behavior of additive mappings defined onR.The first result in this topic is theclassical Posners second theorem.It is proved in 26 that if a prime ringRadmits a nonzeroderivationdsuch
19、that d(x),xZ(R)for allxR,thenRis commutative.This theorem wasregarded as the starting point of many papers concerning the study of various kinds of additivemappings satisfying appropriate algebraic conditions on some subsets of prime and semiprimerings.Over the last few decades,a lot of authors have
20、 investigated the relationship betweenthe commutativity of the ringRand certain specific types of derivations ofR(for instance,see 2 and 11,where further references can be found).In 9,Ashraf and Rehman showedthat ifRis a prime ring,Ia nonzero ideal andda derivation ofRsuch thatd(xy)xyZ(R),d(xy)yxZ(R
21、)ord(x)d(y)xyZ(R)for allx,yI,thenRis commutative.Later,theauthors extended above results to the case of generalized derivations in 10 and generalized(,)-derivations in 18.Recently,in 5,Ali,Dar and Asci continue this line of investigationregarding the commutativity in rings with involution.In fact,th
22、ey proved if a prime ringRwith involutionof a characteristic different from two admits a nonzero derivationdsatisfyingd(xx)xx=0,d(xx)xx=0 ord(x)d(x)xx=0 for allxR,thenRis commutative.Mamouni,Nejjar and Ouhtite 19 explored the commutativity of prime rings with involutionprovided with a generalized de
23、rivation F satisfying any one of the following conditions:(i)F(xx)(xx)Z(R),(ii)F(xx)(xx)Z(R),(iii)F(x)F(x)(xx)Z(R)for all xR.For more related results on this topic,we refer the reader to 21 and 25.Over the past forty years,there has been a great deal of work concerning specified derivationsacting as
24、 a homomorphism or an anti-homomorphism on some distinguished subsets ofR(see 17for a partial bibliography).As is well known,Bell and Kappe in 12 proved the following:(i)ifdis a derivation of a prime ringRwhich acts as a homomorphism or an anti-homomorphism on a nonzero right ideal I of R,then d=0 o
25、n R,(ii)ifdis a derivation of a semiprime prime ringRwhich is either an endomorphism or ananti-endomorphism,then d=0 on R.This result was extended to-derivations in 28 and(,)-derivations in 8.In 6,Albasdiscussed the above identities involving generalized derivations in prime rings which are centralv
26、alued.More precisely,the author proved the following result:LetRbe a prime ring andIbe anonzero ideal ofR,which admits a nonzero generalized derivationFwith associated derivationd of R.(i)IfF(xy)F(x)F(y)Z(R)for allx,yI,thenRis commutative orF=IRorF=IR,(ii)If F(xy)F(y)F(x)Z(R)for all x,yI,then R is c
27、ommutative.No.2HUANG Shu-liang:Differential Identities in Prime Rings with Involution137Recently,in 14,Boua and Ashraf extended Albass results for generalized derivations in primerings with involution.Motivated by the above results,it is natural to question that what can we say about thestructure of
28、Rif we replaced the generalized derivationFby a generalized(,)-derivation.Inthe present paper,we study the above results in the setting of generalized(,)-derivations.Moreover,two examples are provided which shows that the condition in our results is essential.2.Some preliminariesWe shall do a great
29、deal of calculations with Lie product and Jordan product,routinely usethe following basic identities without any specific mention:xy,z,=xy,z,+x,(z)y=xy,(z)+x,z,y,x,yz,=(y)x,z,+x,y,(z),(xo(yz),=(xoy),(z)(y)x,z,=(y)(xoz),+x,y,(z),(xy)oz),=x(yoz),x,(z)y=(xoz),y+xy,(z).We fix the following results which
30、 will be used frequently in the proof of the theorems.Lemma 2.1.(23,Lemma 2.1).LetRbe a prime ring with involution of the second,then is centralizing if and only if R is commutative.Lemma 2.2.(19,Fact 2).Let(R,)be a prime ring of characteristic different from twowith the second involution,then Z(R)T
31、H(R)6=0.Lemma 2.3.(3,Remark 1).LetRbe a prime ring with centerZ(R).Ifa,abZ(R)forsome a,bR,then a=0 or bZ(R).Lemma 2.4.(23,Fact 1).Let(R,)be a 2-torsion free prime ring with involution providedwith a derivationd,thend(h)=0 for allhZ(R)TH(R)implies thatd(z)=0 for allzZ(R).Lemma 2.5.(18,Theorem 4.1).Le
32、tRbe a prime ring andIbe a nonzero ideal ofR.IfRadmits a generalized(,)-derivationF 6=such thatF(xy)(xy)Z(R)for allx,yI,then R is commutative.Lemma 2.6.(18,Lemma 2.7).LetRbe a prime ring andIbe a nonzero right ideal ofR.IfRadmits a generalized(,)-derivationFsuch that06=F(I)Z(R),thenRis commutative.L
33、emma 2.7.(18,Theorem 4.3).LetRbe a prime ring andIa nonzero ideal ofR.IfRadmits a generalized(,)-derivationF 6=such thatF(x)F(y)(xy)Z(R)for allx,yI,then R is commutative.Lemma 2.8.(1,Theorem 2.1).LetRbe a 2-torsion free prime ring,Ja nonzero Jordanideal and a subring ofR.Suppose thatis an automorphi
34、sm ofRandFa generalized(,)-derivation associated with a(,)-derivationd.IfFacts as an anti-homomorphism onJ,theneither d=0 or J Z(R).Lemma 2.9.(27,Theorem 1).LetRbe a 2-torsion free prime ring andUa nonzero square-closed Lie idealR.Suppose thatFandGare two generalized generalized(,)-derivations138CHI
35、NESE QUARTERLY JOURNAL OF MATHEMATICSVol.38associated with(,)-derivationsgandd,respectively.IfG(xy)F(x)F(y)Z(R)for allx,yU,then either d=g=0 or U Z(R).3.Main resultsTheorem 3.1.LetRbe a prime ring of characteristic different from two with involutionof thesecond kind andFa generalized(,)-derivation o
36、fR.IfF 6=andF(xx)(xx)Z(R)for all xR,then R is commutative.Proof.Suppose thatF=0,then we have(xx)Z(R)and also(xx)Z(R),then(x,x)Z(R)for allxR.This implies that x,x1(Z(R)Z(R)for allxR.Con-sequently,by Lemma 2.1,we get the required result.Henceforth,we shall assume thatF 6=0.By the hypothesis,we haveF(x
37、x)(xx)Z(R)for all xR.(3.1)Linearizing(3.1)and using(3.1),we getF(xy)+F(yx)(xy)(yx)Z(R)for all x,yR.(3.2)Sinceis the second kind,Lemma 2.2 yields thatZ(R)TH(R)6=0.For all 06=hZ(R)TH(R),replacing y by yh in(3.2)we finds that(F(xy)+F(yx)(xy)(yx)(h)+(xy+yx)d(h)Z(R)for all x,yR.(3.3)Comparing(3.2)and(3.3
38、)gives that(xy+yx)d(h)Z(R)for allx,yR.This implies that(xy+yx)1d(h)1(Z(R)Z(R)for all x,yR.(3.4)LetD=1d.Then we find thatD(x+y)=1d(x+y)=1d(x)+1d(y)=D(x)+D(y),namely,Dis an additive mapping ofR.Also,D(xy)=1d(xy)=1(d(x)(y)+(x)d(y)=1d(x)y+x1d(y)=D(x)y+xD(y)for allx,yR.This implies thatDis a derivation o
39、fR.Using the fact thatD(Z(R)Z(R),it follows from(3.4)and Lemma 2.3 that eitherxy+yxZ(R)for allx,yRor1d(h)=0 for all 06=hZ(R)TH(R).We have to distinguish twocases.Case 1.Ifxy+yxZ(R)for allx,yR,lettingx=y,then 2xxZ(R)for allxR.sinceRis 2-torsion free,we arrive atxxZ(R)and hencexxZ(R)for allxR.Combinin
40、gthe last two equations,we get x,xZ(R)for allxR.Again using Lemma 2.1,we concludethat R is commutative.Case 2.If1d(h)=0 for all 06=hZ(R)TH(R),then1d(Z(R)=0 by Lemma 2.4.For 06=sZ(R)TS(R)and yR,substituting ys for y in(3.2),we can conclude that(F(xy)+F(yx)+(xy)(yx)(s)Z(R)for all x,yR.(3.5)Calculating
41、 the sum of equations(3.2),(3.5)and using the fact that the characteristic ofRisdifferent from two yields(F(yx)(yx)(s)Z(R)for allx,yR.Using the fact(s)6=0No.2HUANG Shu-liang:Differential Identities in Prime Rings with Involution139and Lemma 2.3,we haveF(yx)(yx)Z(R)for allx,yR.In particular,F(xy)(xy)
42、Z(R)holds for all x,yR.Application of Lemma 2.5,we are done.Theorem 3.2.Let R be a prime ring of characteristic different from two with involution ofthe second kind andFa generalized(,)-derivation ofR.IfF 6=andF(xx)+(xx)Z(R)for all xR,then R is commutative.Proof.IfF(xx)+(xx)Z(R)for allxR,then the ge
43、neralized(,)-derivationFsatisfies the relation(F)(xx)(xx)Z(R)for all xR.Thereby,the proof is completedby Theorem 3.1.Theorem 3.3.LetRbe a prime ring of characteristic different from two with involutionof thesecond kind andFa generalized(,)-derivation ofR.IfF 6=andF(x)F(x)(xx)Z(R)for all xR,then R is
44、 commutative.Proof.We are given thatF(x)F(x)(xx)Z(R)for all xR.(3.6)Replacing x by x+y in(3.6)and using(3.6)yields thatF(x)F(y)+F(y)F(x)(xy+yx)Z(R)for all x,yR.(3.7)For all 06=hZ(R)TH(R),replace y by yh in(3.7)to get(F(x)F(y)+F(y)F(x)(xy+yx)(h)+(F(x)(y)+(y)F(x)d(h)Z(R).(3.8)Combine(3.7)and(3.8)to ge
45、t(F(x)(y)+(y)F(x)d(h)Z(R)for allx,yR.Thisimplies that(1F(x)y+y1F(x)1d(h)Z(R)for allx,yR.By Lemma 2.3,either1F(x)y+y1F(x)Z(R)for all x,yR or 1d(h)=0 for all 06=hZ(R)TH(R).Now suppose that1F(x)y+y1F(x)Z(R)for allx,yR.Take 06=yZ(R)TH(R).The last equation becomes(1F(x)+1F(x)yZ(R)for allx,yR.By virtue of
46、 Lemma 2.3,1F(x)+1F(x)Z(R)for allxR.Thisimplies thatF(x)+F(x)Z(R)for allxR.Similarly,Taking 06=yZ(R)TS(R)we can getF(x)+F(x)Z(R)for allxR.Combining the last two relations and using the fact thatthe characteristic ofRis different from two,we obtain thatF(x)Z(R)for allxR,namely,06=F(R)Z(R).It follows
47、from Lemma 2.6 thatRis commutative.The remaining case is1d(h)=0 for all 06=hZ(R)TH(R).Recalling that1dis a derivation ofR,in view ofLemma 2.4,1d(Z(R)=0 and henced(Z(R)=0 sinceis an automorphism ofR.Now,for06=sZ(R)TS(R),replace y by ys in(3.7)to get(F(x)F(y)+F(y)F(x)+(xyyx)(s)Z(R)for all x,yR.(3.9)Ri
48、ght multiplying(3.7)by(s)gives that(F(x)F(y)+F(y)F(x)(xy+yx)(s)Z(R)for all x,yR.(3.10)140CHINESE QUARTERLY JOURNAL OF MATHEMATICSVol.38Adding(3.9)and(3.10),we are force that(F(y)F(x)(yx)(s)Z(R)for allx,yR.Using the same technique as in the proof of Theorem 3.1,we getF(x)F(y)(xy)Z(R)forall x,yR.Accor
49、dingly,R is commutative by Lemma 2.7.By similar arguments as above with necessary variations,we can prove the following result.Theorem 3.4.LetRbe a prime ring of characteristic different from two with involutionof thesecond kind andFa generalized(,)-derivation ofR.IfF 6=andF(x)F(x)+(xx)Z(R)for all x
50、R,then R is commutative.Theorem 3.5.Let R be a prime ring of characteristic different from two with involution ofthe second kind.IfF 6=is a generalized(,)-derivation associated with a(,)-derivationdof R such that F(xx)F(x)F(x)Z(R)for all xR,then R is commutative or d=0.Proof.We are given thatF(xx)F(