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        Notes on Automorphisms of Prime Rings

        2015-11-03 11:42:35HuangShuliang

        Huang Shu-liang

        (Department of Mathematics,Chuzhou University,Chuzhou,Anhui,239012)

        Communicated by Du Xian-kun

        Notes on Automorphisms of Prime Rings

        Huang Shu-liang

        (Department of Mathematics,Chuzhou University,Chuzhou,Anhui,239012)

        Communicated by Du Xian-kun

        Let R be a prime ring,L a noncentral Lie ideal and σ a nontrivial automorphism of R such that usσ(u)ut=0 for all u∈L,where s,t are fixed non-negative integers.If either charR>s+t or charR=0,then R satisfies s4,the standard identity in four variables.We also examine the identity(σ([x,y])-[x,y])n= 0 for all x,y∈I,where I is a nonzero ideal of R and n is a fixed positive integer.If either charR>n or charR=0,then R is commutative.

        prime ring,Lie ideal,automorphism

        2010 MR subject classification:16N60,16U80,16W25

        Document code:A

        Article ID:1674-5647(2015)03-0193-06

        1 Introduction

        The standard identity s4in four variables is defined as follows:

        where(-1)τis the sign of the permutation τ of the symmetric group of degree 4.In the following,unless stated otherwise,R always denotes a prime ring with its center Z(R)and Martindale quotient ring Q.The center of Q,denoted by C,is called the extended centroid of R(we refer the reader to[1]for these terminologies).For any x,y∈R,the symbol[x,y]stands for the commutator xy-yx.An additive subgroup U of R is said to be a Lie ideal of R if[u,r]∈U for all u∈U and r∈R.For nonempty subsets A,B of R,let[A,B]be the additive subgroup generated by all the elements of the form[a,b]with a∈A and b∈B. Recall that a ring R is prime if for any a,b∈R,aRb=(0)implies a=0 or b=0,and is semiprime if for any a∈R,aRa=(0)implies a=0.An additive mapping d:R→R is called a derivation if d(xy)=d(x)y+xd(y)holds for all x,y∈R.Starting from thisdefinition,Breˇsar[2]first introduced the definition of generalized derivations:An additive mapping F:R→R is called a generalized derivation if there exists a derivation d:R→R such that F(xy)=F(x)y+xd(y)holds for all x,y∈R,and d is called the associated derivation of F.Hence,the concept of generalized derivations covers the concepts of both derivations and left multipliers(i.e.,the additive mappings satisfying F(xy)=F(x)y for all x,y∈R).Basic examples are derivations and generalized inner derivations(i.e.,mappings of the typefor some a,b∈R).We prefer to call such mappings generalized inner derivations for the reason that they present a generalization of the concept of inner derivations(i.e.,mappings of the formfor some a∈R).

        This paper is included in a line of investigation concerning the relationship between the global structure of a ring R and the behaviors of some additive mappings defined on R that satisfy certain special identities.A well-known result of Herstein[3]states that if ρ is a right ideal of R such that un=0 for all u∈ρ,where n is a fixed positive integer,then ρ=0. Chang and Lin[4]considered the situation when d(u)un=0 for all u∈ρ,where d is a nonzero derivation of R.Dhara and De Filippis[5]studied the case when usH(u)ut=0 for all u∈L,where L is a noncommutative Lie ideal of R,H is a generalized derivation of R and s,t are fixed non-negative integers.More precisely,they proved the following:Let R be a prime ring,H a nonzero generalized derivation of R and L a noncommutative Lie ideal of R.Suppose that usH(u)ut=0 for all u∈L.Then R satisfies s4,the standard identity in four variables.On the other hand,Carini and De Filippis[6]proved that if R is a prime ring with charR/=2 and such that[d(u),u]n=0 for all u∈L,where L is a noncentral Lie ideal and d is a nonzero derivation of R,then R is commutative.Wang[7]also discussed the identity[σ(u),u]n=0 replacing the derivation d by an automorphism σ of R and obtained that R satisfies s4.Motivated by the previous results,our first objective in this note is to study the identity usσ(u)ut=0 for all u∈L,where L is a noncentral Lie ideal and σ is an automorphism of the prime ring R,and then describe the structure of R.

        During the past few decades,there has been an ongoing interest in concerning the relationship between the commutativity of a ring and the existence of certain specific types of derivations(see[8-9]for a partial bibliography,where further references can be found).The first result in this direction is due to Posner[10]who proved that a prime ring R admitting a nonzero derivation d such that[d(x),x]∈Z(R)for all x∈R must be commutative.This result was subsequently refined and extended by a number of authors,and we refer the readers to[11-13]for further references.In 1992,Daif and Bell[14]showed that if in a semiprime ring R there exist a nonzero ideal I of R and a derivation d such that d([x,y])=[x,y]for all x,y∈I,then I?Z(R).If R is a prime ring,this implies that R is commutative.De Filippis[15]obtained the commutativity of prime rings when the derivation d is replaced by an automorphism σ.Later,Quadri et al.[16]extended Daif's result to generalized derivations.Ashraf and Ali[17]proved that R is commutative in the setting of left multipliers.In 2002,De Filippis[18]obtained the following result:Let R be a prime ring without non-zero nil right ideal,d a nonzero derivation of R,and I a non-zero ideal of R.If for any x,y∈I,there exists n=n(x,y)≥1 such that(d([x,y])-[x,y])n=0,then R is commutative.Itis natural to ask what we can say about the commutativity of rings satisfying the identity(σ([x,y])-[x,y])n=0 for an automorphism of the ring.Our second object in this paper is to investigate the above identity in prime rings and to obtain the commutativity of prime rings.

        2 Main Results

        Theorem 2.1Let R be a prime ring,L a noncentral Lie ideal and σ a nontrivial automorphism of R such that usσ(u)ut=0 for all u∈L,where s,t are fixed non-negative integers.If either charR>s+t or charR=0,then R satisfies s4,the standard identity in four variables.

        Proof.Suppose that R does not satisfy s4.It follows from pages 4-5 in[3],Lemma 2 in[19]and Theorem 4 in[20]that there exists a nonzero two-sided ideal I of R such that 0/=[I,R]?L.In particular,[I,I]?L.Hence,without loss of generality,we may assume that L=[I,I].By assumption,we have

        which can be rewritten as

        By Main Theorem in[21],we divide the proof into two cases.

        Case 1.Let σ be Q-outer.Since either charR>s+t or charR=0,by Theorem 3 in[22],we obtain

        In particular,let

        Then it follows from Theorem 2 in[23]that R is commutative and so L is central,a contradiction.

        Case 2.If σ is Q-inner,then there exists an invertible element b∈Q such that

        We note that b/∈C since σ/=1R,the identity map on R.By Theorem 2 in[24],I,R and Q satisfy the same generalized polynomial identities(or GPIs in brief),and from(2.1)we have

        In the case that the center C of Q is infinite,we have

        For any given v∈V,we want to show that v and bv are linearly D-dependent.If bv=0,then v and bv are D-dependent.We are done in this case.Suppose that bv/=0,v and bv were D-independent.Then by the density of R,there would exist x,y∈R such that

        From(2.2),we can see that

        It is a contradiction.From above we have proven that

        where αv∈D depends on v∈V.In fact,it is easy to check that αvis independent of the choice of v∈V.Indeed,for any v,w∈V,by the above arguments,there exist αv,αw,αv+w∈D such that

        And so

        Hence

        If v and w are D-independent,then

        and we are done.Otherwise,suppose that v and w were D-dependent.Let v=λw for some λ∈D.Then

        That is,for any v we can choose some αvsuch that αv=αw,where w/=0 and αwwere fixed beforehand.So we conclude that there exists a δ∈D such that bv=vδ for all v∈V. So b=δ∈D=C.It is a contradiction.This completes the proof.

        Theorem 2.2Let R be a prime ring,I a nonzero ideal and σ a nontrivial automorphism of R such that

        where n is a fixed positive integer.If either charR>n or charR=0,then R is commutative. Proof.For n=1,it follows from Lemma 1.2 in[15].Suppose that n≥2 and R were not commutative.We give that

        Assume first that σ is Q-outer.Since either charR>n or charR=0,by Theorem 3 in[22],we see that

        In particular,setting u=0,we have

        It follows from Theorem 2 in[23]that R is commutative,a contradiction.Suppose now that σ is Q-inner.Then there would exist an invertible element b∈Q-C such that

        Suppose that there exists a v∈V such that bv and v are D-independent.Then by the density of R,there exist x,y∈R such that

        Application of(2.3)yields that

        a contradiction.The rest of the proof is the same as the last paragraph of the proof of Theorem 2.1.And hence we can get a contradiction that b∈C.The proof is thereby completed.

        The following example demonstrates that R being prime is essential in Theorem 2.2. Example 2.1Set

        and

        where Z is the ring of all integers.Next,let us define a mapping σ:R→R by

        Then it is clear that I is a nonzero ideal of R and σ is a nontrivial automorphism of R. And it is easy to check that(σ([x,y])-[x,y])n=0 for all x,y∈I.However,R is not commutative.

        References

        [1]Beidar K I,Martindale W S,Mikhalev V.Rings with Generalized Identities.New York:Marcel Dekker Inc.,1996.

        [2]Breˇsar M.On the distance of the composition of two derivations to be the generalized derivations.Glasgow Math.J.,1991,33:89-93.

        [3]Herstein I N.Topics in Ring Theory.Chicago:Univ.of Chicago Press,1969.

        [4]Chang C M,Lin Y C.Derivations on one-sided ideals of prime rings.Tamsui Oxford J.Management Sci.,2001,17(2):139-145.

        [5]Dhara B,De Filippis V.Notes on generalized derivations on Lie ideals in prime rings.Bull. Korean Math.Soc.,2009,46(3):599-605.

        [6]Carini L,De Filippis V.Commutators with power central values on a Lie ideal.Pacific J. Math.,2000,193(2):269-278.

        [7]Wang Y.Power-centralizing automorphisms of Lie ideals in prime rings.Comm.Algebra,2006,34:609-615.

        [8]Bell H E,Martindale W S.Centralizing mappings of semiprime rings.Canad.Math.Bull.,1987,30(1):92-101.

        [9]Demir C,Argac N.A result on generalized derivations with Engel conditions on one-sided ideal.J.Korean Math.Soc.,2010,47(3):483-494.

        [10]Posner E C.Derivations in prime rings.Proc.Amer.Math.Soc.,1957,8:1093-1100.

        [11]Argac N,Carini L,De Filippis V.An engel condition with generalized derivations on Lie ideals. Taiwanese J.Math.,2008,12:419-433.

        [12]De Filippis V.An Engel condition with generalized derivations on multilinear polynomials. Israel J.Math.,2007,162:93-108.

        [13]Lee T K.Left annihilators characterized by GPIs.Trans.Amer.Math.Soc.,1995,347(8):3159-3165.

        [14]Daif M N,Bell H E.Remarks on derivations on semiprime rings.Internat.J.Math.Math. Sci.,1992,15:205-206.

        [15]De Filippis V.Automorphisms and derivations in prime rings.Rend.Mat.Appl.(7),1999,19:393-404.

        [16]Quadri M A,Khan M S,Rehman N.Generalized derivations and commutativity of prime rings.Indian J.Pure Appl.Math.,2003,34(9):1393-1396.

        [17]Ashraf M,Ali S.On left multipliers and the commutativity of prime rings.Demonstratio Math.,2008,41(4):763-771.

        [18]De Filippis V.On a subset with nilpotent values in a prime ring with derivation.Boll.Un. Mat.Ital.B(7),2002,8(5):833-838.

        [19]Di Vincenzo O M.On the n-th centralizers of a Lie ideal.Boll.Un.Mat.Ital.A(7),1989,3:77-85.

        [20]Lanski C,Montgomery S.Lie structure of prime ring of characteristic 2.Pacific J.Math.,1972,42(1):117-136.

        [21]Chuang C L.Differential identities with automorphisms and anti automorphisms I.J.Algebra,1992,149:371-404.

        [22]Chuang C L.Differential identities with automorphisms and anti automorphisms II.J.Algebra,1993,160:130-171.

        [23]Herstein I N.Center-like elements in prime rings.J.Algebra,1979,60:567-574.

        [24]Chuang C L.GPIs having coefficents in Utumi quotient rings.Proc.Amer.Math.Soc.,1988,103(3):723-728.

        [25]Erickson J S,Martindale W S,Osborn J M.Prime nonassociative algebras.Pacific J.Math.,1975,60(1):49-63.

        [26]Martindale W S.Prime rings satisfying a generalized polynomial identity.J.Algebra,1969,12:176-584.

        [27]Jacobson N.Structure of Rings.Provindence,RI,USA:Amer.Math.Soc.,1964.

        10.13447/j.1674-5647.2015.03.01

        date:Sept.16,2011.

        The NSF(1408085QA08)of Anhui Province,the Natural Science Research Foundation(KJ2014A183)of Anhui Provincial Education Department,and Anhui Province College Excellent Young Talents Fund Project(2012SQRL155)of China.

        E-mail address:shulianghuang@163.com(Huang S L).

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