A Note on Posner s Theorem with Generalized Derivations on Lie Ideals

A Note on Posner's Theorem with Generalized Derivations on Lie Ideals

V. DEFILIPPIS(*) - M. S. TAMMAMEL-SAYIAD(**)

ABSTRACT- Let R be a prime ring of characteristic different from 2, Z(R) it s center, U its Utumi quotient ring, C its extended centroid, G a non-zero generalized derivation ofR,La non-central Lie ideal ofR. We prove that if [G(u);u];G(u)

2Z(R) for allu2Lthen one of the following holds:

1. there existsa2Csuch that G(x)ˆax, for allx2R;

2. R satisfies the standard identity s4(x1;. . .;x4) and there exista2U, a2Csuch thatG(x)ˆax‡xa‡ax, for allx2R.

1. Introduction.

The motivation for this paper lies in an attempt to extend in some way the well known first Posner's Theorem contained in [15]: there Posner proved that ifdis a derivation of a prime ringRsuch that [d(x);x] falls in the center ofR, for allx2R, then eitherdˆ0 orRis a commutative ring.

Recently in [3] Cheng studied derivations of prime rings that satisfy cer- tain special Engel type conditions: he showed that ifRis a prime ring of characteristic different from 2 and d a non-zero derivation of R which satisfies the condition [[d(x);x];d(x)]ˆ0 for all x2R, then R mustbe commutative.

Our purpose here is to continue this line of investigation by studying the setSˆ f[[G(x);x];G(x)];x2Lg, whereGis a generalized derivation

(*) Indirizzo dell'A.: DI.S.I.A., Faculty of Engineering University of Messina, Contrada di Dio, 98166, Messina, Italy.

E-mail: defilippis@unime.it

(**) Indirizzo dell'A.: Department of Mathematics, Faculty of Science Beni Suef University, Beni Suef, Egypt.

E-mail: m_s_tammam@yahoo.com

2000Mathematics Subject Classification. 16N60, 16W25.

defined on R andLis a non-central Lie ideal of R. More specifically an additive mapG:R !Ris said to be a generalized derivation if there is a derivation d of R such that, for all x;y2R, G(xy)ˆG(x)y‡xd(y). A significative example is a map of the form G(x)ˆax‡xb, for some a;b2R; such generalized derivations are called inner. Our goal is to confirm that there is a relationship between the structure of the prime ring Rand the behaviour of suitable additive mappings defined onRthat satisfy certain special identities. We will show that if any element ofSis central in R, then some informations about the form of the generalized derivationG and the structure ofRcan be obtained. More precisely we will prove the following:

THEOREM. Let R be a prime ring of characteristic different from 2, Z(R)its center, U its Utumi quotient ring, C its extended centroid, G a non-zero generalized derivation of R, L a non-central Lie ideal of R. We prove that if

[G(u);u];G(u)

2Z(R)for all u2L then one of the following holds:

1. there existsa2C such that G(x)ˆax, for all x2R;

2. R satisfies the standard identity s4(x1;. . .;x4)and there exist a2U, a2C such that G(x)ˆax‡xa‡ax, for all x2R.

For sake of clearness we premitthe following:

FACT1. Denote byTˆUCCfXgthe free product overCof theC- algebraUand the freeC-algebraCfXg, withXa coutable set consisting of non-commuting indeterminates fx1;. . .;xn;. . .g. The elements of T are called generalized polynomial with coefficients inU. Moreover ifIis a non- zero ideal ofR, thenI;RandUsatisfy the same generalized polynomial identities with coefficients inU. For more details about these objects we refer the reader to [1] and [4].

FACT2. Let a₁;. . .;a_k2U be linearly independentover C and

a₁g₁(x₁;. . .;x_n)‡. . .‡a_kg_k(x₁;. . .;x_n)ˆ02T, for some g₁;. . .;g_k2 2TˆU_CCfXg. As a consequence of the result in [4], if for any i, g_i(x1;. . .;xn)ˆPⁿ

jˆ1x_jh_j(x1;. . .;xn) and h_j(x1;. . .;xn)2T, t hen g₁(x₁;. . .;x_n);. . .;g_k(x₁;. . .;x_n) are the zero element of T. The same conclusion holds if g₁(x₁;. . .;x_n)a₁‡. . .‡g_k(x₁;. . .;x_n)a_kˆ02T, and g_i(x₁;. . .;x_n)ˆPⁿ

jˆ1h_j(x₁;. . .;x_n)x_jfor someh_j(x₁;. . .;x_n)2T.

2. The case of Inner Generalized Derivations.

In this section we study the case when the generalized derivationGis inner defined as follows:G(x)ˆax‡xbfor allx2R, wherea;bare fixed elements ofU.

In all that follows we denote P(x1;x2;x3)ˆh

a[x1;x2]‡[x1;x2]b;[x1;x2]

;a[x1;x2]‡[x1;x2]bi x3

x3h

a[x1;x2]‡[x1;x2]b;[x1;x2]

;a[x1;x2]‡[x1;x2]bi and assume thatRsatisfies the generalized identityP(x₁;x₂;x₃).

LEMMA1. If R does not satisfy any non trivial generalized polynomial identity, then a;b2C and G(x)ˆax, for all x2R and foraˆa‡b.

PROOF. Denote by TˆU_CCfx₁;x₂;x₃g the free product overC of theC-algebraUand the freeC-algebraCfx₁;x₂;x₃g. Any elementofTis a generalized polynomial with coefficients inU.

Suppose thatRdoes not satisfy any non trivial generalized polynomial identity. Thus

P(x1;x2;x3)ˆh

a[x1;x2]²‡[x1;x2](b a)[x1;x2] [x1;x2]²b;a[x1;x2]‡[x1;x2]b

x3

x3

ha[x1;x2]²‡[x1;x2](b a)[x1;x2] [x1;x2]²b;a[x1;x2]‡[x1;x2]bi

ˆa

[x1;x2]²a[x1;x2]‡[x1;x2]³b [x1;x2]a[x1;x2]² [x₁;x₂]²(b a)[x₁;x₂]‡[x₁;x₂]³b

x₃

‡[x1;x2]

(b a)[x1;x2]a[x1;x2]‡(b a)[x1;x2]²b [x1;x2]ba[x1;x2] [x1;x2]b[x1;x2]b ba[x1;x2]² b[x1;x2](b a)[x1;x2]‡b[x1;x2]²b

x3

x3

ha[x1;x2]²‡[x1;x2](b a)[x1;x2] [x1;x2]²b;a[x1;x2]‡[x1;x2]bi

ˆ02T:

Suppose thatf1;agare linearlyC-independent. SinceP(x₁;x₂;x₃) is a tri- vial generalized polynomial identity forR, then

[x1;x2]²a[x1;x2]‡[x1;x2]³b [x1;x2]a[x1;x2]² [x1;x2]²(b a)[x1;x2]

‡[x1;x2]³b

x3ˆ02T

that is

[x1;x2]²a[x1;x2]‡[x1;x2]³b [x1;x2]a[x1;x2]² [x₁;x₂]²(b a)[x₁;x₂]‡[x₁;x₂]³bˆ02T:

This implies thatf1;bgare linearlyC-dependent. In fact, if not it fol- lows that [x1;x2]³is an identity forR, a contradiction. Thusbˆb2Cand Rsatisfies

[x1;x2]²a[x1;x2]‡b[x1;x2]³ [x1;x2]a[x1;x2]²‡[x1;x2]²a[x1;x2] which is a non-trivial generalized identity, since we suppose thatf1;agare linearlyC-independent. This contradiction says thatf1;agare linearlyC- dependent, that isaˆa2C.

ThereforeRsatisfies the generalized identity

h[x1;x2](b‡a)[x1;x2] [x1;x2]²(b‡a);[x1;x2](b‡a)i x3

x3

h[x1;x2](b‡a)[x1;x2] [x1;x2]²(b‡a);[x1;x2](b‡a)i that is

0ˆ

[x₁;x₂]b⁰[x₁;x₂]²b⁰ [x₁;x₂]²b⁰[x₁;x₂]b⁰ [x₁;x₂]b⁰[x₁;x₂]b⁰[x₁;x₂]

‡[x1;x2]b⁰[x1;x2]²b⁰ x3

x3

[x1;x2]b⁰[x1;x2]²b⁰ [x1;x2]²b⁰[x1;x2]b⁰ [x1;x2]b⁰[x1;x2]b⁰[x1;x2]

‡[x₁;x₂]b⁰[x₁;x₂]²b⁰

ˆ

[x₁;x₂]b⁰[x₁;x₂]²b⁰ [x₁;x₂]²b⁰[x₁;x₂]b⁰ [x₁;x₂]b⁰[x₁;x₂]b⁰[x₁;x₂]

‡[x1;x2]b⁰[x1;x2]²b⁰ x3

x3[x1;x2]b⁰[x1;x2]b⁰[x1;x2] x3

2[x1;x2]b⁰[x1;x2]² [x1;x2]²b⁰[x1;x2] b⁰

whereb⁰ˆb‡a. Iff1;b⁰gare linearlyC-independent, then x₃

2[x₁;x₂]b⁰[x₁;x₂]² [x₁;x₂]²b⁰[x₁;x₂]

is a non-trivial generalized identity forR, a contradiction. Thenf1;b⁰gare linearlyC-dependent, that isb⁰2Cas well asb, and we are done. p LEMMA2. Let RˆMm(F)be the ring of mm matrices over the field F of characteristic different from2, with m>1, a;b elements of R such that

[au‡ub;u];au‡ub 2Z(R)

for all u2[R;R]. Then one of the following holds:

1) a;b2Z(R);

2) a b2Z(R)andmˆ2.

PROOF. The first aim is to prove that a bis a diagonal matrix. Say aˆP

ij a_ije_ij, bˆP

ij b_ije_ij, where a_ij;b_ij2F, and e_ij are the usual matrix units. Letuˆ[r₁;r₂]ˆ[e_ii;e_ij]ˆe_ij, for anyi6ˆj. Thus

[aeij‡eijb;eij];aeij‡eijb

ˆ(bji aji)(aji bji)eij2Z(R) that is all the off-diagonal entries of the matrixa bare zeros.

Letnow x2Aut_F(R) with x(x)ˆ(1‡e_ji)x(1 e_ji). Of course [x(a)u‡ux(b);u];x(a)u‡ux(b)

2Z(R), for allu2[R;R]. By calculation we have that

x(a)ˆa‡e_jia ae_ji e_jiae_ji x(b)ˆb‡e_jib be_ji e_jibe_ji

and by the previous argument we also have that x(a b) is a diagonal matrix. In particular the (j;i)-entry of x(a b) is zero, tha is a_ii b_iiˆa_jj b_jj. By the arbitrariness ofi6ˆj, we have thata bˆais a central matrix in R and

[au‡ua‡au;u];au‡ua‡au

2Z(R), for all u2[R;R], that isRsatisfies

ha[x₁;x₂]² [x₁;x₂]²a;a[x₁;x₂]‡[x₁;x₂]a‡a[x₁;x₂]

;x₃i : In casemˆ2 we are done. Thus assume thatm3.

Suppose thatais nota diagonal matrix, for example letaji6ˆ0 fori6ˆj.

Letnowvˆ[e_ii;e_ij‡e_ji]ˆe_ij e_ji. Thus Xˆ

av² v²a;av‡va‡av 2Z(R)

hence for anyk6ˆi;j, the (k;i)-entry X_ki of the matrixX is zero. By cal- culations we have that

Xkiˆaki(aij aji)‡akj(ajj‡aii‡a)ˆ0 (1)

On the other hand forwˆ[e_ii;e_ij e_ji]ˆe_ij‡e_ji we have Yˆ

aw² w²a;aw‡wa‡aw

2Z(R)ˆ0:

Again the (k;i)-entryY_kiof the matrixY is zero, that is Y_kiˆa_ki(a_ij‡a_ji)‡a_kj(a_jj‡a_ii‡a)ˆ0 (2)

By (1) and (2) itfollows that

2a_kia_jiˆ0 (3)

Therefore we have that ifa_ji6ˆ0, thena_kiˆ0 for allk6ˆi;j.

Let W2Aut_F(R) defined as W(x)ˆ(1‡e_kj)x(1 e_kj). Of course for all u2[R;R],

W(a)u² u²W(a);W(a)u‡uW(a)‡au

2Z(R) Since the (k;i)- entry of the matrixW(a) is equal toaji6ˆ0, then by the argument in (3) we have that the (j;i)-entrya⁰_ji of the matrix W(a) is zero. By calculations it follows 0ˆa⁰_jiˆa_ji, a contradiction. Therefore a mustbe a diagonal matrix in R. As above, for all r6ˆs, let x2Aut_F(R) with x(x)ˆ

ˆ(1‡e_sr)x(1 e_sr). Hence alsox(a)ˆa‡e_sra ae_sr e_srae_sr mustbe a diagonal matrix. In particular the (s;r)-entry ofx(a) is zero, tha isarrˆass. By the arbitrariness ofi6ˆj, we have thatais a central matrix inR, and we

are done again. p

PROPOSITION1. LetRbe a prime ring of characteristic different from2.

Suppose thata; bare elements ofUsuch that

[au‡ub; u]; au‡bu

2Z(R),

for allu2[R; R]. Then one of the following holds:

1) a;b2C;

2) a b2CandRsatisfies the standard identitys₄(x₁;. . .;x₄).

PROOF. By Lemma 1 we may assume thatRsatisfies the non-trivial generalized polynomial identity

P(x1;x2;x3)ˆh

a[x1;x2]²‡[x1;x2](b a)[x1;x2] [x1;x2]²b;a[x1;x2]‡[x1;x2]bi x3

x3

ha[x1;x2]²‡[x1;x2](b a)[x1;x2] [x1;x2]²b;a[x1;x2]‡[x1;x2]bi : By a theorem due to Beidar (Theorem 2 in [1]) this generalized polynomial identity is also satisfied byU. In caseCis infinite, we haveP(r1;r2;r3)2Cfor all r1;r2;r32UN

CC, whereC is the algebraic closure ofC. Since both U and UN

CCare centrally closed ([6], Theorems 2.5 and 3.5), we may replaceRbyUor UN

CCaccording asCis finite or infinite. Thus we may assume thatRis cen- trally closed overCwhich is either finite or algebraically closed. By Martindale's theorem [14],Ris a primitive ring having a non-zero socleHwithCas the as- sociated division ring. In light of Jacobson's theorem ([10], pag 75) R is iso- morphic to a dense ring of linear transformations on some vector spaceVoverC.

Assume firstthatVis finite-dimensional overC. Then the density ofR onVimplies thatRM_k(C), the ring of allkkmatrices overC. SinceR is not commutative we assumek2. In this case the conclusion follows by Lemma 2.

Assume nextthatV is infinite-dimensional over C. As in lemma 2 in [16], the set [R;R] is dense on Rand so from P(r1;r2;r3)2Z(R), for all r1;r2;r32R, we have

[ar‡rb;r];ar‡rb

2Z(R), for all r2R. Due to the infinity-dimensionality,Rcannot satisfies any polynomial identity. In particular the non-zero idealHcannotsatisfiess₄(x₁;. . .;x₄). Suppose that eithera2=Corb2=C, then at least one of them doesn't centralize the non zero idealHofR, and we will prove that this leads to a contradiction.

Hence we are supposing that there exist h1;h22H such that either [a;h₁]6ˆ0 or [b;h₂]6ˆ0 and there exist h₃;h₄;h₅;h₆2H such that s₄(h₃;. . .;h₆)6ˆ0.

Lete²ˆeany non-trivial idempotentelementofH. Forrˆexe, with anyx2R, we have that

[aexe‡exeb;exe];aexe‡bexe

2Z(R). By com- muting with (1 e) and then right multiplying by (1 e) itfollows 2(1 e)a(exe)³b(1 e)ˆ0. Since char(R)6ˆ2, we have that either (1 e)aeˆ0 or eb(1 e)ˆ0. If (1 e)aeˆ0 then aeˆeae and baeˆbeae. On the other hand, in caseeb(1 e)ˆ0, we getebˆebe, and soebaˆabea. In any case we notice that the ringeResatisfies the gen- eralized identityh

[(eae)X‡X(ebe);X];(eae)X‡X(ebe)

;Yi .

By Litoff's theorem in [7] there exists e²ˆe2H such that h₁;h₂, h₃;h₄;h₅;h₆2eRe, moreover eRe is a central simple algebra finite di- mensional over its center. Sinces₄(h₃;. . .;h₆)6ˆ0, then eReM_t(C), for t3. By the finite dimensional case, we have thateae;ebe2Z(eRe), but this contradicts with the choices ofh1;h2 ineRe. p

3. The proof of the Theorem.

In this final section we will make use of the result of Kharchenko [11]

about the differential identities on a prime ring R. We refer to Chapter 7 in [2] for a complete and detailed description of the theory of generalized polynomial identities involving derivations.

Itis well known thatany derivation of a prime ring R can be uniquely extended to a derivation of its Utumi quotients ring U, and so any deri- vation ofRcan be defined on the wholeU ([2], pg. 87).

Now, we denote by Der(Q) the set of all derivations on Q. By a deri- vation word we mean an additive mapDof the formDˆd1d2. . .dm, with each di2Der(Q). Then a differential polynomial is a generalized poly- nomial, with coefficients in Q, of the form F(D_j(x_i)) involving non- commutative indeterminates x_i on which the derivations wordsD_j actas

unary operations. The differential polynomialF(Dj(xi)) is said a differential identity on a subsetTofQif itvanishes for any assignmentof values from Tto its indeterminatesx_i.

LetD_int be the C-subspace of Der(Q) consisting of all inner deriva- tions on Qand letd andd be two non-zero derivations on R. As a par- ticular case of Theorem 2 in [11] we have the following result (see also Theorem 1 in [13]):

FACT3. Ifdis a non-zero derivation onRandF(x1;. . .;xn;d(x1);. . .;d(xn)) is a differential identity onR, then one of the following holds:

1) eitherd2D_int;

2) orRsatisfies the generalized polynomial identity F(x1;. . .;xn;y1;. . .;yn;):

Now we are ready to prove our main result:

THEOREM. Let R be a prime ring of characteristic different from2, Z(R)its center, U its Utumi quotient ring, C its extended centroid, G a non-zero generalized derivation of R, L a non-central Lie ideal of R. We prove that if

[G(u);u];G(u)

2Z(R)for all u2L then one of the following holds:

1) there existsa2Csuch thatG(x)ˆax, for allx2R;

2) R satisfies the standard identity s4(x1;. . .;x4) and there exist a2U,a2Csuch thatG(x)ˆax‡xa‡ax, for allx2R.

PROOF. By Theorem 3 in [12] every generalized derivation g on a dense rightideal ofR can be uniquely extended to the Utumi quotient ringUofR, and thus we can think of any generalized derivation ofRto be defined on the whole U and t o be of t he form g(x)ˆax‡d(x) for some a2U and d a derivation on U. Thus we will assume in all that follows that there exist a2U and d derivation on U such that G(x)ˆax‡d(x). We note that we may assume that R is notcommu- tative, sinceLis notcentral. Moreover, since char(R)6ˆ2, there exists a non-central two-sided idealIofRsuch that [I;I]L(see p. 4-5 in [8];

Lemma 2 and Proposition 1 in [5]). Therefore [[G(u);u];G(u)]2Z(R) for all u2[I;I]. Moreover by [13] R and I satisfy the same differ- ential polynomial identities, that is [[G(u);u];G(u)]2Z(R) for all u2[R;R].

By assumptionRsatisfies the differential identity (4) h

[a[x1;x2]‡[d(x1);x2]‡[x1;d(x2)];[x1;x2]];a[x1;x2]‡[d(x1);x2]‡[x1;d(x2)]i x3

x3

h[a[x1;x2]‡[d(x1);x2]‡[x1;d(x2)];[x1;x2]];a[x1;x2]‡[d(x1);x2]‡[x1;d(x2)]i

Firstsuppose thatdis notan inner derivation onU. By Kharchenko's theorem [11]Rsatisfies the polynomial identity

(5) h

a[x1;x2]‡[y1;x2]‡[x1;y2];[x1;x2]

;a[x1;x2]‡[y1;x2]‡[x1;y2]i x3

x3h

a[x1;x2]‡[y1;x2]‡[x1;y2];[x1;x2]

;a[x1;x2]‡[y1;x2]‡[x1;y2]i in particularRsatisfies any blended component

ha[x₁;x₂];[x₁;x₂]

;a[x₁;x₂]i

x₃ x₃h

a[x₁;x₂];[x₁;x₂]

;a[x₁;x₂]i

that is h

a[x1;x2];[x1;x2]

;a[x1;x2]i 2Z(R)

and by Proposition 1 we have thataˆa2C. Thus from (5), itfollows thatR satisfies the polynomial identity

h[y₁;x₂]‡[x₁;y₂];[x₁;x₂]

;a[x₁;x₂]‡[y₁;x₂]‡[x₁;y₂]i x₃ x3h

[y1;x2]‡[x1;y2];[x1;x2]

;a[x1;x2]‡[y1;x2]‡[x1;y2]i : SinceRsatisfies a polynomial identity, there existsM_k(F), the ring of all matrices over a suitable field F, such thatRandM_k(F) satisfy the same polynomial identities (see [9], Theorem 2 p.54 and Lemma 1 p.89). For x₁ˆe₂₂,x₂ˆe₂₁,y₁ˆe₂₁andy₂ˆe₁₂we obtain

[y1;x2]ˆ0; [x1;y2]ˆ e12; [x1;x2]ˆe21

and it follows the contradiction h e₁₂;e₂₁

;ae₂₁ e₁₂i

ˆ2e₁₂‡2ae₂₁2=Z(R):

Now consider the case whendis an inner derivation induced by the ele- ment b2U. Since G(x)ˆax‡[b;x]ˆax‡bx xbˆ(a‡b)x‡x( b) andbyProposition1,wehavethat eithera;b2Cora‡2b2CandRsatisfies

s4(x1;. . .;x4). In the first case we conclude thatG(x)ˆax, fora2C; in t he

second oneG(x)ˆa⁰x‡xa⁰‡ax, wherea⁰ˆ b. p

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