Right Sided Ideals and Multilinear Polynomials with Derivation on Prime Rings

Right Sided Ideals and Multilinear Polynomials with Derivations on Prime Rings

BASUDEBDHARA(*) - RAJENDRAK. SHARMA(**)

ABSTRACT- Le tRbean associativeprimering of charR6ˆ2with centerZ(R) and extended centroidC,f(x1;. . .;xn) a nonzero multilinear polynomial overCinn noncommuting variables,da nonzero derivation ofRandra nonzero right ideal of R. Weprovethat: (i) if [d²(f(x1;. . .;xn));f(x1;. . .;xn)]ˆ0for allx1;. . .;xn2r thenrCˆeRCfor some idempotent elementein thesocleofRCandf(x1;. . .;xn) is central-valued ineRCeunlessdis an inner derivation induced byb2Qsuch that b²ˆ0 and brˆ0; (ii) if [d²(f(x1;. . .;xn));f(x1;. . .;xn)]2Z(R) for all

x1;. . .;xn2rthenrCˆeRCfor some idempotent elementein thesocleofRC

and eitherf(x1;. . .;xn) is central ineRCeoreRCesatisfies the standard identity S4(x1;x2;x3;x4) unlessdis an inner derivation induced byb2Qsuch thatb²ˆ0 andbrˆ0.

Throughout this paper,Ralways denotes a prime ring with extended centroidCandQits two-sided Martindale ring of quotient. Bydwemean a nonzero derivation ofR. Forx;y2R, thecommutator ofx;yis denoted by [x;y] and defined by [x;y]ˆxy yx. Wedenote[x;y]₂ˆ[[x;y];y]ˆ

ˆ[x;y]y y[x;y].

A well known result proved by Posner [17] states that R must be commutativeif [d(x);x]2Z(R) for all x2R. In [10] Lanski generalized the Posner's result to a Lie ideal. More precisely Lanski proved that ifL is a noncommutativeLieideal ofRsuch that [d(x);x]2Z(R) for allx2L,

(*) Indirizzo dell'A.: Department of Mathematics, Belda College, Belda, Paschim Medinipur-721424, India.

E-mail: basu dhara@yahoo.com

(**) Indirizzo dell'A.: Department of Mathematics, Indian Institute of Technol- ogy, Delhi, Hauz Khas, New Delhi-110016, India.

E-mail: rksharma@maths.iitd.ac.in

Mathematics Subject Classification: 16W25, 16R50, 16N60.

This work is supported by a grant from University Grants Commission, India.

then char Rˆ2 and R satisfies S4(x1;x2;x3;x4), thestandard identity.

Notethat a noncommutativeLieideal ofR contains all thecommutators [x₁;x₂] forx₁;x₂ in some nonzero ideal ofR( see [10, Lemma 2 (i), (ii)]).

So, it is natural to consider the situation when [d(x);x]2Z(R) for all commutators xˆ[x₁;x₂] or more general case xˆf(x₁;. . .;x_n) where

f(x1;. . .;xn) is a multilinear polynomial. In [11] Lee and Lee proved that

if [d(f(x1;. . .;xn));f(x1;. . .;xn)]2Z(R) for allx1;. . .;xn in somenonzero ideal of R, then f(x1;. . .;xn) is central-valued on R, except when char Rˆ2 and Rsatisfies S₄(x₁;x₂;x₃;x₄). Recently, De Filippis and Di Vin- cenzo (see [7]) consider the situationd([d(f(x₁;. . .;x_n));f(x₁;. . .;x_n)])ˆ0 for all x₁;. . .;x_n2R, where d and d aretwo derivations of R. The statement of De Filippis and Di Vincenzo's theorem is the following:

THEOREMA ([7, Theorem 1]). Let K be a noncommutative ring with unity, R a prime K-algebra of characteristic different from 2, d and d nonzero derivations of R and f(x₁;. . .;x_n) a multilinear polynomial over K. If d([d(f(x1;. . .;xn));f(x1;. . .;xn)])ˆ0 for all x1;. . .;xn2R, then f(x1;. . .;xn) is central-valued on R.

In casedand d are two same derivations, the differential identity be- comes [d²(f(x₁;. . .;x_n));f(x₁;. . .;x_n)]ˆ0 for all x₁;. . .;x_n2R. So, it is natural to ask, what happen in cases[d²(f(x₁;. . .;x_n));f(x₁;. . .;x_n)]2Z(R) for all x1;. . .;xn2R and [d²(f(x1;. . .;xn));f(x1;. . .;xn)]2Z(R) for all

x1;. . .;xn 2r, whereris a non-zero right ideal of R. In the present paper our

object is to study these cases.

For the sake of completeness we recall some basic notations, de- finitions and some easy consequences of the result of Kharchenko [8]

about the differential identities on a prime ring R. First, wedenoteby Der(Q) theset of all derivations on Q. By a derivation word D of R we

mean Dˆd₁d₂d₃. . .d_m for somederivations d_i of R. For x2R, we

denote by x^D theimageof xunder D, that isx^Dˆ( (x^d1)^d2 )^dm. By a differential polynomial, we mean a generalized polynomial, with coefficients in Q, of the form F(x^D_i^j) involving noncommutativein- determinates x_i on which thederivations words D_j act as unary op- erations. F(x^D_i^j)ˆ0 is said to be a differential identity on a subset Tof Q if it vanishes for any assignment of values from T to its in- determinates x_i.

Now letDint betheC-subspaceofDer(Q) consisting of all inner deri- vations onQ. By Kharchenko's theorem [8, Theorem 2], we have the fol- lowing result:

LetRbe a prime ring of characteristic different from 2. If two nonzero derivationsdanddareC-linearly independent moduloDintandF(x^D_i^j) is a differential identity onR, whereD_jarederivations words of thefollowing formd;d;d²;dd;d², thenF(y_ji) is a generalized polynomial identity onR, wherey_jiare distinct indeterminates.

As a particular case, we have:

If d is a nonzero derivation on R and F(x1;. . .;xn;x^d₁;. . .;x^d_n; x^d₁²;. . .;x^d_n²) is a differential identity onR, then one of the following holds:

(i) eitherd2Dint

or

(ii) R satisfies the generalized polynomial identity F(x₁;. . .;x_n; y₁;. . .;y_n;z₁;. . .;z_n)

Denote byQCCfX1;. . .;Xngthefreeproduct of theC-algebraQand

CfX1;. . .;Xng, thefreeC-algebra in noncommuting indeterminates

X1;. . .;Xn.

Sincef(x₁;. . .;x_n) is a multilinear polynomial, we can write f(x₁;. . .;x_n)ˆx₁x₂. . .x_n‡ X

I6ˆs2Sn

a_sx_s(1). . .x_s(n)

where S_n is the permutation group over n elements and any a_s2C.

Wedenoteby f^d(x1;. . .;xn) thepolynomial obtained from f(x1;. . .;xn) by replacing each coefficient as with d(as:1). In this way wehave

d(f(x1;. . .;xn))ˆf^d(x1;. . .;xn)‡X

i

f(x1;. . .;d(x_i);. . .;xn) and

d²(f(x1;. . .;xn))ˆd(f^d(x1;. . .;xn))‡dX

i

f(x1;. . .;d(xi);. . .;xn)

ˆf^d2(x₁;. . .;x_n)‡X

i

f^d(x₁;. . .;d(x_i);. . .;x_n)

‡X

i

f^d(x1;. . .;d(x_i);. . .;xn)‡X

i6ˆj

f(x1;. . .;d(x_i);. . .;d(x_j);. . .;xn)

‡X

i

f(x₁;. . .;d²(x_i);. . .;x_n)

ˆf^d2(x₁;. . .;x_n)‡2X

i

f^d(x₁;. . .;d(x_i);. . .;x_n)

‡2X

i5j

f(x1;. . .;d(xi);. . .;d(xj);. . .;xn)‡X

i

f(x1;. . .;d²(xi);. . .;xn):

1. The case forrˆR.

LEMMA1.1. Let RˆM_k(F)be the ring of all kk matrices over a field F of characteristic 6ˆ2, b2R and f(x₁;. . .;x_n) is a multilinear poly- nomial over F. If k2and[[b;[b;f(x1;. . .;xn)]];f(x1;. . .;xn)]ˆ0for all x1;. . .;xn2R or if k3 and [[b;[b;f(x1;. . .;xn)]];f(x1;. . .;xn)]2Z(R) for all x1;. . .;xn2R, then either b2FI_k or f(x1;. . .;xn)is central-va- lued on R.

PROOF. Le tbˆ(b_ij)_kk. Le te_ijbetheusual matrix unit with 1 in (i;j) entry and zero else where. Now we proceed to show that b2Z(R) if 2f(x1;. . .;xn) is non central valued onR.

For simplicity, wewritef(x₁;. . .;x_n)ˆf(x), where xˆ(x₁;. . .;x_n) RⁿˆR R(ntimes). Then by assumption,

[[b;[b;f(x)]];f(x)]ˆ[b²f(x) 2bf(x)b‡f(x)b²;f(x)]2Z(R) for all x2Rⁿ. Since f(x1;. . .;xn) is assumed to be noncentral on R, by [15, Lemma 2, Proof of Lemma 3] there exists a sequence of matrices

rˆ(r₁;. . .;r_n) in R such that f(r)ˆf(r₁;. . .;r_n)ˆae_ij6ˆ0 where

06ˆa2F and i6ˆj. Thus

[b²ae_ij 2bae_ijb‡ae_ijb²;ae_ij]2Z(R):

Sincetherank of [b²aeij 2baeijb‡aeijb²;aeij] is 2, [b²aeij 2baeijb‡

‡aeijb²;aeij]ˆ0. Left multiplying by eij, we get 0ˆeij( 2baeijbaeij)ˆ

ˆ 2a²b²_jie_ij. SincecharF6ˆ2,b_jiˆ0. Fors6ˆt, le tsbea permutation in thesymmetric group S_m such that s(i)ˆs and s(j)ˆt. Le t c bethe automorphism of R defined by x^cˆP

p;qj_pqe_pqc

ˆP

p;qj_pqe_s(p);s(q). Then

f(r^c)ˆf(r₁^c;. . .;r_n^c)ˆf(r)^c ˆaest6ˆ0 and wehaveas abovebtsˆ0 for

s6ˆt. Thusbis a diagonal matrix. For anyF-automorphismuofR,b^uenjoys thesameproperty as b does, namely, [[b^u;[b^u;f(x)]];f(x)]2Z(R) for all x2Rⁿ. Hence,b^umust bediagonal. WritebˆP^k

iˆ1a_iie_ii; then for eachj6ˆ1, wehave

(1‡e1j)b(1 e1j)ˆX^k

iˆ1

aiieii‡(bjj b11)e1j

diagonal. Therefore,b_jjˆb11and sobis a scalar matrix.

LEMMA1.2. Let R be a prime ring of characteristic different from2 and f(x1;. . .;xn) a multilinear polynomial over C. If for any iˆ1;. . .;n,

[f(x₁;. . .;z_i;. . .;x_n);f(x₁;. . .;x_n)]ˆ0

for all x₁;. . .;x_n;z_i2R, then the polynomial f(x₁;. . .;x_n)is central-valued on R.

PROOF. Le t a be a noncentral element ofR. Then replacingz_i with [a;x_i] wehavethat for anyiˆ1;. . .;n

[f(x₁;. . .;[a;x_i];. . .;x_n);f(x₁;. . .;x_n)]ˆ0 and so

"

Xⁿ

iˆ0

f(x1;. . .;[a;xi];. . .;xn);f(x1;. . .;xn)

#

ˆ0

which implies, [a;f(x1;. . .;xn)]2ˆ0 for allx1;. . .;xn2R. By [11, Theorem], f(x1;. . .;xn) is central-valued onR.

THEOREM 1.3. Let R be a prime ring of characteristic different from 2, d a nonzero derivation of R, f(x₁;. . .;x_n) a multilinear polynomial over C. If

[d²(f(x1;. . .;xn));f(x1;. . .;xn)]2Z(R) for allx1;. . .;xn2R;

then either f(x₁;. . .;x_n)is central-valued on R or R satisfies the standard identity S₄(x₁;x₂;x₃;x₄).

PROOF. Le t I be any nonzero two-sided ideal of R. If for every r₁;. . .;r_n 2I, [d²(f(r₁;. . .;r_n));f(r₁;. . .;r_n)]ˆ0, then by [14], this generalized differential identity is also satisfied by Q and hence by R as well. By Theorem A, f(r1;. . .;rn) is then central-valued on R and wearedone. Now weassumethat for somer₁;. . .;rn 2I, 06ˆ[d²(f(r₁;. . .;r_n)); f(r₁;. . .;r_n)]2I\Z(R). Thus I\Z(R)6ˆ0. Let K be a nonzero two-sided ideal of R_Z, the ring of central quotients of R.

Since K\R is a nonzero two-sided ideal of R, (K\R)\Z(R)6ˆ0.

Therefore, K contains an invertible element in RZ and so RZ is a simplering with identity 1.

By assumption,Rsatisfies the differential identity g(x₁;. . .;x_n;d(x₁);. . .;d(x_n);d²(x₁);. . .;d²(x_n))

ˆ[[f^d2(x₁;. . .;x_n)‡2X

i

f^d(x₁;. . .;d(x_i);. . .;x_n)

‡2X

i5j

f(x1;. . .;d(x_i);. . .;d(x_j);. . .;xn)

‡X

i

f(x1;. . .;d²(xi);. . .;xn);f(x1;. . .;xn)];xn‡1]:

Ifdis notQ-inner, then by Kharchenko's theorem [8],

"

f^d2(x1;. . .;xn)‡2X

i

f^d(x1;. . .;yi;. . .;xn)

‡2X

i5j

f(x1;. . .;yi;. . .;yj;. . .;xn)

‡X

i

f(x₁;. . .;z_i;. . .;x_n);f(x₁;. . .;x_n)

;x_n‡1

#

ˆ0 (1)

for all xi;yi;zi;xn‡12R for iˆ1;2;. . .;n. In particular, for any i, as- sumingy1ˆ ˆyi 1ˆyi‡1ˆ ˆynˆ0;z1ˆ ˆznˆ0, wehave

[[f^d2(x₁;. . .;x_n)‡2f^d(x₁;. . .;y_i;. . .;x_n);f(x₁;. . .;x_n)];x_n‡1]ˆ0 and so

"

f^d2(x1;. . .;xn)‡2X

i

f^d(x1;. . .;yi;. . .;xn);f(x1;. . .;xn)

;xn‡1

#

ˆ0 for allx_i;y_i;xn‡12R,iˆ1;2;. . .;n. Thus from (1), weobtain

"

2X

i5j

f(x₁;. . .;y_i;. . .;y_j;. . .;x_n)

‡X

i

f(x1;. . .;zi;. . .;xn);f(x1;. . .;xn)

;xn‡1

#

ˆ0 (2)

for allx_i;y_i;z_i;xn‡12Rforiˆ1;2;. . .;n.

By localizingRatZ(R), weobtain that (2) is also an identity ofR_Z. Since RandR_Zsatisfy the same polynomial identities, in order to prove thatR satisfiesS₄, wemay assumethatRis a simplering with 1. ThusRsatisfies theidentity (2). Now putting yiˆ[b;xi]ˆd(xi) and ziˆ[b;[b;xi]]ˆ

ˆd²(xi),iˆ1;2;. . .;n for someb2=Z(R), where d is an inner derivation induced by someb2R, weobtain thatRsatisfies

[[d²(f(x1;. . .;xn));f(x1;. . .;xn)];xn‡1]ˆ0:

Thus by Martindale's theorem [16], Ris a primitivering with a minimal right ideal, whose commuting ringDis a division ring which is finitedi- mensional overZ(R). However, sinceRis simplewith 1,Rmust beArti- nian. HenceRˆD_k⁰, thering ofk⁰k⁰ matrices overD, for somek⁰1.

Again, by [9, Lemma 2], it follows that there exists a field F such that RM_k(F), thering of all kkmatrices over the fieldF, andM_k(F) sa- tisfies

[[d²(f(x1;. . .;xn));f(x1;. . .;xn)];xn‡1]ˆ0:

Ifk3, then by Lemma 1.1, we haveb2Z(R), a contradiction. Thuskˆ2, that is,RsatisfiesS₄(x₁;x₂;x₃;x₄).

Similarly, thesameconclusion can bedrawn in cased is an Q-inner derivation induced by someb2Q.

2. The case for one-sided ideal.

We begin with the following lemmas

LEMMA2.1. Letrbe a nonzero right ideal of R and d a derivation of R.

Then the following conditions are equivalent:

(i) d is an inner derivation induced by some b2Q such that brˆ0;

(ii) d(r)rˆ0.

For its proof, we refer to [2, Lemma].

LEMMA 2.2. Let R be a prime ring, r a nonzero right ideal of R,

f(x₁;. . .;x_t)a multilinear polynomial over C, a2R and n a fixed positive

integer. If f(x₁;. . .;x_t)ⁿaˆ0 for all x₁;. . .;x_t2r, then either aˆ0 or f(r)rˆ0.

For its proof, we refer to [3, Lemma 2 (II)].

LEMMA2.3. Let R be a prime ring. If[d²(f(x1;. . .;xn));f(x1;. . .;xn)]2 Z(R) for all x₁;. . .;x_n2r, then R satisfies nontrivial generalized polynomial identity unless d is an inner derivation induced by b2Q such that b²ˆ0and brˆ0.

PROOF. Supposeon thecontrary thatRdoes not satisfy any nontrivial generalized polynomial identity (GPI). Thus we may assume that R is noncommutative, otherwiseRsatisfies trivially a nontrivial GPI. Now we consider the following two cases:

CASEI. Supposethatdis aQ-inner derivation induced by an element b2Qsuch thatb²6ˆ0. Then for anyx02r

[[b;[b; f(x₀X₁;. . .;x₀X_n)]]; f(x₀X₁;. . .;x₀X_n)]2Z(R) that is

[[b²f(x₀X₁;. . .;x₀X_n) 2bf(x₀X₁;. . .;x₀X_n)b

‡f(x₀X₁;. . .;x₀Xn)b²; f(x₀X₁;. . .;x₀Xn)];x₀X_n‡1] (3)

is a GPI for R, so it is the zero element in Q_CCfX₁;. . .;X_n‡1g.

Denote l_R(r) theleft annihilator of r inR. Supposefirst that f1;b;b²g arelinearly C-independent modulo lR(r), that is (ab²‡bb‡g)rˆ0 if and only if aˆbˆgˆ0. Since R is not a GPI-ring, a fortiori it can not be a PI-ring. Thus, by [13, Lemma 3] there exists x02r such that fb²x₀;bx₀;x₀g arelinearly C-independent. Then we have that

[[b²f(x0X1;. . .;x0Xn) 2bf(x0X1;. . .;x0Xn)b

‡f(x₀X₁;. . .;x₀X_n)b²;f(x₀X₁;. . .;x₀X_n)];x₀X_n‡1]ˆ0 is a nontrivial GPI forR, a contradiction.

Therefore, f1;b;b²g arelinearly C-dependent modulo lR(r), that is there exista;b;g2C, not all zero, such that (ab²‡bb‡g)rˆ0. Suppose that aˆ0. Then b6ˆ0, otherwise gˆ0. Thus by (bb‡g)rˆ0, wehave that (b‡b ¹g)rˆ0. Sincebandb‡b ¹ginduce the same inner derivation, wemay replaceb by b‡b ¹g in the basic hypothesis. Therefore, in any casewemay supposebrˆ0 and then from (3), R satisfies

x₀X_n‡1f²(x₀X₁;. . .;x₀X_n)b²ˆ0. Since R does not satisfy any nontrivial

GPI, b²ˆ0, a contradiction.

Next suppose that a6ˆ0. In this casethereexist l;m2C such that b²x0ˆlbx0‡mx0 for all x02r. If bx0 and x0 arelinearly C- dependent for all x02r, then again we obtain brˆ0 and so b²ˆ0.

Therefore choose x₀2r such that bx₀ and x₀ arelinearly C-in- dependent. Then replacing b²x₀ with lbx₀‡mx₀, weobtain from (3)

that R satisfies

h(lb‡m)f²(x0X1;. . .;x0Xn) 2bf(x0X1;. . .;x0Xn)bf(x0X1;. . .;x0Xn)

‡f(x₀X₁;. . .;x₀X_n)(lb‡m)f(x₀X₁;. . .;x₀X_n)g f(x0X1;. . .;x0Xn)(lb‡m)f(x0X1;. . .;x0Xn)

2f(x₀X₁;. . .;x₀X_n)bf(x₀X₁;. . .;x₀X_n)b‡f²(x₀X₁;. . .;x₀X_n)b² ;x₀X_n‡1i : This is a nontrivial GPI forR, because the term

lbf²(x0X1;. . .;x0Xn) 2bf(x0X1;. . .;x0Xn)bf(x0X1;. . .;x0Xn) x0Xn‡1

appears nontrivially, a contradiction.

CASEII. Supposethatdis an inner derivation induced by an elementb2Q such that b²ˆ0. Thus wehavethat [ 2bf(X₁;. . .;X_n)b;f(X₁;. . .;X_n)]2 Z(R) is satisfied by r. In casethereexists x02r such that fbx0;x0g arelinearly C-independent, we have that [[ 2bf(x0X1;. . .;x0Xn)b;

f(x₀X₁;. . .;x₀X_n)];x₀X_n‡1] is a non trivial GPI forR, a contradiction. Hence fbx₀;x₀garelinearlyC-dependent for allx₀2r, that is there existsa2C such that (b a)rˆ0. Thus wehavethat [af²(X1;. . .;Xn)(a b);Xn‡1] is satisfied byr, in particularRsatisfies:

[af²(x₀X₁;. . .;x₀X_n)(a b);f(x₀X₁;. . .;x₀X_n)]ˆaf³(X₁;. . .;X_n)(a b) for anyx02r. SinceRis not GPI, it follows that eitherbˆa2C, which is a contradiction, oraˆ0 which meansbrˆ0, as required.

CASEIII. Supposethatdis an inner derivation induced by an element b2Q such that brˆ0. Thus wehavethat [ f²(X1;. . .;Xn)b²;Xn‡1] is satisfied byr, in particularRsatisfies:

[ f²(x0X1;. . .;x0Xn)b²;f(x0X1;. . .;x0Xn)]ˆf³(x0X1;. . .;x0Xn)b² for anyx₀2r. Again sinceRis not GPI weconcludethatb²ˆ0.

CASE IV. Next suppose that d is not Q-inner derivation. By our as- sumption wehavethatRsatisfies

0ˆh

f^d2(xX1;. . .;xXn)‡2P

i f^d(xX1;. . .;d(x)Xi‡xd(Xi);. . .;xXn)

‡2P

i5j f(xX1;. . .;d(x)Xi‡xd(Xi);. . .;d(x)Xj‡xd(Xj);. . .;xXn)

‡P

i f(xX1;. . .;d²(x)X_i‡2d(x)d(X_i)‡xd²(X_i);. . .;xXn);f(xX1;. . .;xXn)

;Xn‡1

i:

By Kharchenko's theorem [8], hf^d2(xX1;. . .;xXn)‡2P

i f^d(xX1;. . .;d(x)X_i‡xr_i;. . .;xXn)

‡2P

i5j f(xX1;. . .;d(x)Xi‡xri;. . .;d(x)Xj‡xrj;. . .;xXn)

‡P

i f(xX₁;. . .;d²(x)X_i‡2d(x)r_i‡xs_i;. . .;xX_n);f(xX₁;. . .;xX_n)

;X_n‡1i

ˆ0 for all X1;. . .;Xn;r1;. . .;rn;s1;. . .;sn2R. In particular, for r1ˆ

ˆr2ˆ ˆrnˆ0, wehave hf^d2(xX1;. . .;xXn)‡2P

i f^d(xX1;. . .;d(x)X_i;. . .;xXn) 2P

i5jf(xX1;. . .;d(x)Xi;. . .;d(x)Xj;. . .;xXn)‡P

i f(xX1;. . .;d²(x)Xi;. . .;xXn)‡

‡P

i f(xX1;. . .;xsi;. . .;xXn);f(xX1;. . .;xXn)

;Xn‡1

iˆ0:

HenceRsatisfies the blended component

[[f(xs₁;. . .;xX_n);f(xX₁;. . .;xX_n)];X_n‡1]ˆ0 which is a nontrivial GPI forR, a contradiction.

THEOREM2.4. Let R be an associative prime ring of char R6ˆ2with center Z(R)and extended centroid C, f(x1;. . .;xn)a nonzero multilinear polynomial over C in n noncommuting variables, d a nonzero derivation of R andra nonzero right ideal of R. If[d²(f(x1;. . .;xn));f(x1;. . .;xn)]ˆ0 for all x₁;. . .;x_n 2rthenrCˆeRC for some idempotent e in the socle of RC and f(x₁;. . .;x_n)is central-valued on eRCe unless d is an inner deri- vation induced by b2Q such that b²ˆ0and brˆ0.

PROOF. Supposedis not aQ-inner derivation induced by an element b2Qsuch thatb²ˆ0 andbrˆ0.

Now assumefirst that f(r)rˆ0, that is f(x₁;. . .;x_n)x_n‡1ˆ0 for all

x1;x2;. . .;xn‡12r. Then by [12, Proposition], rCˆeRC for someidem-

potent e2soc(RC). Since f(r)rˆ0, wehavef(rR)rRˆ0 and hence f(rQ)rQˆ0 by [4, Theorem 2]. In particular,f(rC)rCˆ0, or equivalently, f(eRC)eˆ0. Thenf(eRCe)ˆ0, that is,f(x₁;. . .;x_n) is a PI foreRCeand, a fortiori, central valued oneRCe.

Next assume thatf(r)r6ˆ0, that isf(x1;. . .;xn)xn‡1is not an identity forrand then we derive a contradiction. By Lemma 2.3,Ris a GPI-ring

and so is also Q (see [1] and [4]). By [16], Q is a primitivering with Hˆsoc(Q)6ˆ0. Moreover, we may assumef(rH)rH6ˆ0, otherwise by [1]

and [4], f(rQ)rQˆ0, which is a contradiction. Choosea₀;a₁;. . .;an2rH such thatf(a₁;. . .;a_n)a₀6ˆ0. Leta2rH. SinceHis a regular ring, there existse²ˆe2Hsuch that

eHˆaH‡a0H‡a1H‡ ‡anH:

Then e2rH and aˆea;a_iˆea_i for iˆ0;1;. . .;n. Thus wehave

f(eHe)ˆf(eH)e6ˆ0. By our assumption and by [14, Theorem 2], we also assumethat

[d²(f(x1;. . .;xn));f(x1;. . .;xn)]

is an identity for rQ. In particular [d²(f(x₁;. . .;x_n));f(x₁;. . .;x_n)] is an identity forrHand so foreH. It follows that, for allr₁;. . .;r_n2H,

0ˆ[d²(f(er₁;. . .;er_n));f(er₁;. . .;er_n)]:

Wemay writef(x₁;. . .;x_n)ˆt(x₁;. . .;x_{n 1})x_n‡h(x₁;. . .;x_n), where x_n never appears as last variable in any monomials ofh. Le tr2H. Then re- placingrnwithr(1 e), wehave

0ˆ[d²(t(er₁;. . .;er_{n 1})er(1 e));t(er₁;. . .;er_{n 1})er(1 e)]:

(4)

Now, weknow thefact thatd(x(1 e))eˆ x(1 e)d(e) and (1 e)d(ex)ˆ

ˆ(1 e)d(e)exand so

(1 e)d²(ex(1 e))eˆ(1 e)d

d(e)ex(1 e)‡ed(ex(1 e)) e

ˆ(1 e)d(e)d(ex(1 e))e‡(1 e)d(e)d(ex(1 e))e

ˆ 2(1 e)d(e)ex(1 e)d(e):

Thus using this facts, wehavefrom (4),

0ˆ(1 e)[d²(t(er₁;. . .;er_{n 1})er(1 e));t(er₁;. . .;er_{n 1})er(1 e)]

ˆ(1 e)d²(t(er1;. . .;ern 1)er(1 e))t(er1;. . .;ern 1)er(1 e)

ˆ 2(1 e)d(e)t(er₁;. . .;er_{n 1})er(1 e)d(e)t(er₁;. . .;er_{n 1})er(1 e)

ˆ 2((1 e)d(e)t(er1;. . .;ern 1)er)²(1 e):

This implies

0ˆ 2

(1 e)d(e)t(er1;. . .;ern 1)er ³ that is

0ˆ 2

(1 e)d(e)t(er1;. . .;ern 1)eH ³:

By [6], (1 e)d(e)t(er1;. . .;ern 1)eHˆ0 which implies (1 e)d(e)t(er1e;. . .;ern 1e)ˆ0:

SinceeHeis a simpleArtinian ring and t(eHe)6ˆ0 is invariant under the action of all inner automorphisms ofeHe, by [5, Lemma 2],(1 e)d(e)ˆ0 and so d(e)ˆed(e)2eH. Thus d(eH)d(e)H‡ed(H)eHrH and d(a)ˆd(ea)2d(eH)rH. Therefore,d(rH)rH. Denote the left anni- hilator ofrHinHbylH(rH). ThenrHˆ rH

rH\lH(rH), a primeC-algebra with thederivationdsuch thatd(x)ˆd(x), for allx2rH. By assumption, wehavethat

[d²(f(x1;. . .;xn));f(x1;. . .;xn)]ˆ0

for allx₁;. . .;x_n2rH. By TheoremA, eitherdˆ0 orf(x₁;. . .;x_n) is cen- tral-valued onrH.

Ifdˆ0, then d(rH)rHˆ0 and so d(r)rˆ0. By Lemma 2.1,d is an inner derivation induced by an elementb2Qsuch thatbrˆ0. Then for all

x1;. . .;xn2r, wehaveby assumption that

0ˆ[[b;[b;f(x₁;. . .;x_n)]];f(x₁;. . .;x_n)]ˆ f²(x₁;. . .;x_n)b²: By [3, Lemma 4], either b²ˆ0 orf(r)rˆ0. In both cases we have con- tradiction.

Iff(x₁;. . .;x_n) is central-valued onrH, thenrH, as well asr, satisfies

[f(x₁;. . .;x_n);x_n‡1]x_n‡2ˆ0. ThenrCˆeRCfor some idempotent element

e2soc(RC) by [12, Proposition] andf(x1;. . .;xn) is central-valued oneRCe and wearedone.

THEOREM2.5. Let R be an associative prime ring of char R6ˆ2with center Z(R)and extended centroid C, f(x₁;. . .;x_n)a nonzero multilinear polynomial over C in n noncommuting variables, d a nonzero derivation of R andra nonzero right ideal of R. If[d²(f(x1;. . .;xn));f(x1;. . .;xn)]2Z(R) for all x1;. . .;xn2rthenrCˆeRC for some idempotent e in the socle of RC and either f(x₁;. . .;x_n) is central-valued on eRCe or eRCe satisfies S₄(x₁;x₂;x₃;x₄)unless d is an inner derivation induced by b2Q such that b²ˆ0and brˆ0.

PROOF. Supposedis not aQ-inner derivation induced by an element b2Qsuch thatb²ˆ0 andbrˆ0.

If [f(r);r]rˆ0, that is [f(x₁;. . .;x_n);x_n‡1]x_n‡2ˆ0 for all

x1;x2;. . .;xn‡22r, then by [12, Proposition], rCˆeRC for someidem- potente2soc(RC) andf(x1;. . .;xn) is central-valued oneRCe.

So, assumethat [f(r);r]r6ˆ0, that is [f(x₁;. . .;xn);x_n‡1]x_n‡2is not an identity forrand then we derive thateRCesatisfiesS₄. By Lemma 2.3,Ris a GPI-ring and so is alsoQ(see [1] and [4]). By [16],Qis a primitivering with Hˆsoc(Q)6ˆ0. Moreover, we may assume [f(rH);rH]rH6ˆ0, otherwise by [1] and [4], [f(rQ);rQ]rQˆ0, which is a contradiction.

Choosea1;. . .;an‡2;b1;. . .;b52rHsuch that [f(a1;. . .;an);an‡1]an‡26ˆ0 and S₄(b₁;b₂;b₃;b₄)b₅6ˆ0. Let a2rH. Since H is a regular ring, there existse²ˆe2Hsuch that

eHˆaH‡a1H‡ ‡an‡2H‡b1H‡ ‡b5H:

Then e2rH and aˆea;a_iˆea_i for iˆ1;. . .;n‡2, b_iˆeb_i for

iˆ1;. . .;5. Thus wehavef(eHe)ˆf(eH)e6ˆ0. Moreover, by [14, Theo-

rem 2], we may also assume that

[[d²(f(x₁;. . .;x_n));f(x₁;. . .;x_n)];x_n‡1]

is an identity forrQ. In particular, [[d²(f(x1;. . .;xn));f(x1;. . .;xn)];xn‡1] is an identity forrHand so foreH. It follows that, for allr1;. . .;rn‡12H,

0ˆ[[d²(f(er₁;. . .;er_n));f(er₁;. . .;er_n)];er_n‡1]:

Wemay writef(x1;. . .;xn)ˆt(x1;. . .;xn 1)xn‡h(x1;. . .;xn), where xn

never appears as last variable in any monomials ofh. Le tr2H. Then re- placingr_nwithr(1 e) andr_n‡1withr_n‡1(1 e), wehave

(5) 0ˆ[[d²(t(er1;. . .;ern 1)er(1 e));t(er1;. . .;ern 1)er(1 e)];ern‡1(1 e)]:

Now, weknow thefact that d(x(1 e))eˆ x(1 e)d(e), (1 e)d(ex)ˆ

ˆ(1 e)d(e)exand (1 e)d²(ex(1 e))eˆ 2(1 e)d(e)ex(1 e)d(e). Thus using these facts, we have from (5),

0ˆ[[d²(t(er₁;. . .;er_{n 1})er(1 e));t(er₁;. . .;er_{n 1})er(1 e)];er_n‡1(1 e)]

ˆ[d²(t(er₁;. . .;er_{n 1})er(1 e));t(er₁;. . .;er_{n 1})er(1 e)]er_n‡1(1 e) ern‡1(1 e)[d²(t(er1;. . .;ern 1)er(1 e));t(er1;. . .;ern 1)er(1 e)]

ˆ t(er1;. . .;ern 1)er(1 e)d²(t(er1;. . .;ern 1)er(1 e))ern‡1(1 e) er_n‡1(1 e)d²(t(er₁;. . .;er_{n 1})er(1 e))t(er₁;. . .;er_{n 1})er(1 e)

ˆt(er1;. . .;ern 1)er(1 e)d(e)t(er1;. . .;ern 1)er(1 e)d(e)rn‡1(1 e)

‡er_n‡1(1 e)d(e)t(er₁;. . .;er_{n 1})er(1 e)d(e)t(er₁;. . .;er_{n 1})er(1 e):

Replacingrn‡1witht(er1;. . .;ern 1)erin theaboverelation, weget 2t(er1;. . .;ern 1)er((1 e)d(e)t(er1;. . .;ern 1)er)²(1 e)ˆ0:

This implies

2((1 e)d(e)t(er1;. . .;ern 1)er)⁴ˆ0 that is

2

(1 e)d(e)t(er1;. . .;ern 1)eH ⁴ˆ0:

By [6], (1 e)d(e)t(er1;. . .;ern 1)eHˆ0 which implies (1 e)d(e)t(er1e;. . .;ern 1e)ˆ0:

SinceeHeis a simpleArtinian ring and t(eHe)6ˆ0 is invariant under the action of all inner automorphisms ofeHe, by [5, Lemma 2],(1 e)d(e)ˆ0 and so d(e)ˆed(e)2eH. Thus d(eH)d(e)H‡ed(H)eHrH and d(a)ˆd(ea)2d(eH)rH. Therefore,d(rH)rH. Denote the left anni- hilator ofrHinHbylH(rH). ThenrHˆ rH

rH\l_H(rH), a primeC-algebra with thederivationdsuch thatd(x)ˆd(x), for allx2rH. By assumption, wehavethat

[[d²f(x1;. . .;xn);f(x1;. . .;xn)];xn‡1]]ˆ0

for all x₁;. . .;x_n2rH. By Theorem 1.3, either dˆ0 or f(x₁;. . .;x_n) is central-valued onrHorrHsatisfies the standard identityS₄(x₁;. . .;x₄).

Ifdˆ0, then as in the proof of Theorem 2.4, we have d(r)rˆ0 and hence by Lemma 2.1,dis an inner derivation induced by an elementb2Q such thatbrˆ0. Thus for allr1;. . .;rn2rH,

[d²(f(r₁;. . .;r_n));f(r₁;. . .;r_n)]ˆ f(r₁;. . .;r_n)²b²2C:

Commuting both sides withf(r₁;. . .;r_n), weobtainf(r₁;. . .;r_n)³b²ˆ0. In this caseby Lemma 2.2, sinceb²6ˆ0,f(rH)rHˆ0. If f(rH)rHˆ0, then [f(rH);rH]rHˆ0, a contradiction.

Iff(x₁;. . .;x_n) is central-valued onrH, then we obtain that

[f(x₁;. . .;x_n);x_n‡1]x_n‡2 is an identity forr, a contradiction.

Finally, if S4(x1;. . .;x4) is an identity for rH, S4(x1;. . .;x4)x5 is an identity forrHand so forrCand this contradicts the choices of the ele- mentsb₁;. . .;b₅ 2rH. Therefore, we conclude that in any caserCsatisfies a polynomial identity, hence by [12, Proposition], there exists an idempo- tente2Soc(RC) such thatrCˆeRC, as desired.

Acknowledgments. The authors would like to thank the referee for his/

her valuable comments and suggestions to modify some arguments of this paper.

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Manoscritto pervenuto in redazione il 16 marzo 2008.