Generalized Functional Identities with (Anti-)Automorphisms and Derivations on Prime Rings, I

Journal of Algebra 215, 644᎐665 Ž1999. Article ID jabr.1998.7751, available online at http:rrwww.idealibrary.com on

Generalized Functional Identities with Ž Anti-. Automorphisms and Derivations on Prime Rings, I* K. I. Beidar Department of Mathematics, National Cheng-Kung Uni¨ ersity, Tainan, Taiwan

Matej Bresar ˇ Uni¨ ersity of Maribor, Maribor, Slo¨ enia

and M. A. Chebotar Department of Mechanics and Mathematics, Moscow State Uni¨ ersity, Moscow, Russia Communicated by Susan Montgomery Received February 12, 1998

The concept of a generalized functional identity ŽGFI. with Žanti.automorphisms and derivations is a generalization of the notion of a generalized polynomial identity ŽGPI. with Žanti.automorphisms and derivations. In the present paper we show that either a prime ring is GPI or such GFIs have only the ‘‘obvious’’ solutions. Our results simultaneously generalize the results of Martindale, Rowen, Lanski, Kharchenko, and Chuang on GPIs and a number of earlier results on GFIs. 䊚 1999 Academic Press

1. INTRODUCTION We refer the reader to the book of Beidar et al. w5x for the basic terminology and results of the theory of rings with generalized polynomial identities Ži.e., rings with GPIs. involving Žanti.automorphisms and derivations. * The second author was partially supported by a grant from the Ministry of Science of Slovenia. 644 0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.

GENERALIZED FUNCTIONAL IDENTITIES

645

The study of generalized functional identities on prime rings was initiated by the second author in w7x where he proved that either a prime ring is GPI or every generalized functional identity of degree two involving additive maps has only the ‘‘obvious’’ solutions Žthat is, all these maps must be of a very particular form.. Simultaneously he described functional identities ŽFIs. of degree two involving additive maps w8x. Further developments came quickly. The third author obtained analogous results for GFIs of degree m involving Ž m y 1.-additive functions w12x. He also considered GFIs with a single anti-automorphism w13x. In a subsequent paper the first author described FIs of degree m on a Lie ideal of a prime ring involving Ž m y 1.-additive maps w1x. After that in a joint paper of the first author with Martindale w3x functional identities with involution were investigated. In particular, it was shown that a modification of proofs given in w1x allows one to drop the assumption on Ž m y 1.-additivity of the maps involved. In w2x the first and the third authors described GFIs of degree m on Lie ideals of semiprime rings involving arbitrary maps. Finally, in a joint paper of the second and third authors a study of more complicated FIs was initiated and it was shown that a prime ring R satisfies an FI of the form f Ž x . xg Ž x . s 0, where f and g are nonzero additive maps, if and only if the central closure of R is a primitive ring with nonzero socle and the associated division ring is commutative w10x. In the present paper we study GFIs with Žanti.automorphisms and derivations of degree m involving arbitrary maps. Our results simultaneously generalize most of previously known results on GFIs and some results of Martindale, Rowen, Lanski, Kharchenko, and Chuang on rings with generalized polynomial identities Žsee w5x.. We refer the reader to w4, 6, 9᎐11, 14x for applications of the theory of GFIs and FIs to different questions in ring theory. Finally we note that the second author was informed by C.-L. Chuang that he had investigated independently GFIs with Žanti.automorphisms and derivations of degree two and obtained a generalization of w7, Main Theoremx in a somewhat different direction than is done in the present paper. In what follows R is a prime ring with maximal right Žleft. ring of quotients Q m r s Q m r Ž R . Žrespectively Q m l s Q m l Ž R .., extended centroid C and symmetric ring of quotients Q s s Q s Ž R . Žsee w16; 5, Chap. 2x for details.. Recall that C is a field and the C-subalgebra R c s RC : Q s is called the central closure of R. Let Q g  Q m r , Q m l 4 . Let X be an infinite set, C² X : the free algebra with unity on X, and Q² X : the free product of C-algebras Q and C² X :. An element pŽ x 1 , x 2 , . . . , x n . g Q² X : is called a generalized polynomial identity on R if pŽ r 1 , r 2 , . . . , rn . s 0 for all r 1 , r 2 , . . . , rn g R. If R has a nonzero generalized polynomial identity, then we shall say that R is GPI. The structure of prime GPI rings was described by Martindale in w16, Theorem 3x Žsee also w5, Corollary 6.3.3x..

646

BEIDAR, BRESAR ˇ , AND CHEBOTAR

THEOREM 1.1 w5, Corollary 6.3.3x. Let R be a prime ring. Suppose that R satisfies a generalized polynomial identity. Then R c , the central closure of R, contains a nonzero idempotent e such that eR c is a minimal right ideal Ž hence R c is a primiti¨ e ring with nonzero socle . and eR c e is a finite dimensional di¨ ision algebra o¨ er C. Following w5x we denote by G s GŽ R . s AutŽ R . j AntiautŽ R . the group of all automorphisms and antiautomorphisms of R, by Gi s Gi Ž R . the subgroup of all X-inner automorphisms, and by G 0 Ž R . a set of representatives of G modulo Gi . Let EndŽ Q s . be the ring of all endomorphisms of the additive group Q s . Clearly it is a right vector space over C with respect to the multiplication Ž ␾ , c . ¬ ␾ c where q ␾ c s cq ␾ for all q g Q s , ␾ g EndŽ Q s ., and c g C. Let DerŽ R . be the set of all derivations of R. It is well known that every derivation of R can be uniquely extended to a derivation of Q s and so we can consider DerŽ R . as a Lie subring of EndŽ Q s . w5, Proposition 2.5.1x. Let Di be the set of all X-inner derivations of Q s and DŽ R . s ŽDerŽ R ..C q Di . Clearly DŽ R . is a subspace of the right vector space EndŽ Q s . over C. Now let U be the universal enveloping algebra of the Žrestricted. differential Lie algebra DŽ R . Žsee w5, Chap. 5x for details.. Choose a well-ordered right C-basis Bi of Di and a wellordered right C-basis B0 for DŽ R . modulo Di . Taking B0 - Bi , we see that B s B0 j Bi is a well-ordered right C-basis of DŽ R .. Let W be the right C-basis of U relative to Ž B, -. as provided by w5, Theorem 5.3.6 and Theorem 5.4.5x. Let ⌬ s ␦ 1 ␦ 2 ⭈⭈⭈ ␦n g W where each ␦ i g B. Then Ža. ␦ 1 F ␦ 2 F ⭈⭈⭈ F ␦n . Žb. If ␦ k s ␦ kq1 s ⭈⭈⭈ s ␦ kqsy1 and charŽ R . s p ) 0, then s - p. We denote by W 0 the set of all elements of W whose factor lie in B0 and also including 1 Žsee w5, Sect. 5.5x for further details.. Next, we denote by T the skew group ring U A G w5, Sect. 7.1x. Clearly W G s  wg < w g W , g g G 4 is a right C-basis of T. We set T0 s W 0 G 0 . A typical element ⍀ g T0 is of the form ⌬ g where ⌬ g W 0 and g g G 0 . Now form the Cartesian product of X and T0 , which we denote by X T 0 s  x ⍀ < x g X , ⍀ g T0 4 Ž x 1 will be written as x .. We then form the free product Q² X T 0 : of C-algebras Q and C² X T 0 :. An element pŽ x 1 , x 2 , . . . , x n . g Q² X T 0 : is called a reduced T ⬘-identity on R if pŽ r 1 , r 2 , . . . , rn . s 0 for all r 1 , r 2 , . . . , rn g R w5, Sect. 7.8x. According to w5, Theorem 7.8.4x, a prime ring with nonzero reduced T ⬘-identity is GPI.

647

GENERALIZED FUNCTIONAL IDENTITIES

Let m be a positive integer and r 1 , r 2 , . . . , rm g R. We set rm s Ž r 1 , r 2 , . . . , rm .. Of course, rmy1 s Ž r 1 , r 2 , . . . , rmy1 .. Now let F: R my 1 ª Q and 1 F i F m. We define a map F i : R m ª Q as F i Ž rm . s F Ž r 1 , . . . , riy1 , riq1 , . . . , rm .

for all r 1 , . . . , rm g R.

Next, let 1 F i - j F m q 1. We define a map F i j: R mq 1 ª Q by the rule F i j Ž rmq 1 . s F Ž r 1 , . . . , riy1 , riq1 , . . . , r jy1 , r jq1 , . . . , rmq1 . for all r 1 , . . . , rmq1 g R. We set F ji s F i j. The map F is called Ž m y 1.additive if it is additive with respect to each argument. Finally, we define 0-additive mappings as constants. Given a positive integer r, we denote by MapŽ R r , Q . the set of all maps r R ª Q. Clearly MapŽ R r , Q . is an abelian group under the pointwise addition. Moreover, the Q-Q-bimodule structure on Q induces a Q-Qbimodule structure on MapŽ R r , Q .. Further, the pointwise multiplication of functions induces a Q-Q-bimodule map Map Ž R r , Q . mQ Map Ž R s , Q . ª Map Ž R rqs , Q . , Ž F m G .Ž a rqs . s F Ž a r .GŽ a rq1 , . . . , a rqs . for all F g MapŽ R r , Q ., G g MapŽ R s, Q . and a rqs g R rqs. We denote by FG the image of F m G under this map. Every element f Ž x r . g Q² X T 0 : defines a map f g MapŽ R r , Q . by the rule a r ¬ f Ž a r ., a r g R r Žsee w5, Chap. 7x for details.. Given Ei jk , Fl st g MapŽ R r , Q ., a ikj, btl s g Q and ␴j g T0 , we can now form a map r

H Ž xr . s

u

ni j

Ý Ý Ý

Eii jk Ž x r . x i␴ j a ikj q

is1 js1 ks1

r

u

mls

Ý Ý Ý btl s x l␴ Fllst Ž x r . s

ls1 ss1 ts1

g Map Ž R r , Q . . We shall say that H is a generalized functional identity Žbrifly, GFI. on R if there exists a finite dimensional subspace V of the vector space Q over C such that H Ž a r . g V for all a r g R r. Clearly, the concept of a GFI is a generalization of that of a GPI. If 1 F i F r, t g R, p, q g Q, and H1Ž x r ., H2 Ž x r . are GFIs on R, then pH1 Ž x r . q H2 Ž x r . q, H1 Ž x 1 , . . . , x iy1 , x i t , x iq1 , . . . , x r . are GFIs on R as well. Our goal is to prove the following result.

BEIDAR, BRESAR ˇ , AND CHEBOTAR

648

THEOREM 1.2. Let R be a prime ring, Q g  Q m r , Q m l 4 , let r G 1 and u G 1 be integers, let ␴j g T0 , j s 1, 2, . . . , u, be distinct elements, and let n i j , m i j , i s 1, 2, . . . , r, j s 1, 2, . . . , u, be nonnegati¨ e integers. Further, let Ei jk , Fl st : R ry1 ª Q, 1 F i, l F r, 1 F j, s F u, 1 F k F n i j , 1 F t F m l s , be maps and let V be a finite dimensional subspace of the ¨ ector space Q o¨ er C. Suppose that r

u

ni j

r

u

ml s

Ý Ý Ý Eii jk Ž x r . x i␴ aikj q Ý Ý Ý btl s x l␴ Fllst Ž x r . g V j

s

is1 js1 ks1

Ž 1.

ls1 ss1 ts1

for all x r g R r , where  a1i j, . . . , a ni ji j 4 and  b1l s, . . . , bml sl s 4 are C-independent subsets of Q for all i, l s 1, 2, . . . , r, j, s s 1, . . . , u, for which n i j / 0 and m l s / 0 Ž n i j s 0 should be understood as that the summand in Ž1. corresponding to x i␴ j does not appear .. Then either R is GPI, or there exist unique maps pi jk l st : R ry2 ª Q, and ␭ i jk t : R ry1 ª C such that u

Eii jk Ž x r . s

ml s

Ý Ý Ý

mi j l btl s x l␴ s pii jk l st Ž x r . q

1FlFr ss1 ts1 l/i u

Fllst

Ý ␭ii jk t Ž x r . bti j , ts1

ni j

nls

Ž x r . s y Ý Ý Ý pii jkl l st Ž x r . x i␴ aikj y Ý ␭ll sk t Ž x r . alks j

1FiFr js1 ks1 i/l

ks1

for all x r g R r , 1 F i, l F r, 1 F j, s F u, 1 F k F n i j , 1 F t F m l s . Further, r

u

ni j

Ý Ý Ý is1 js1 ks1

Eii jk

Ž xr .

x i␴ j a ki j

r

q

u

mls

Ý Ý Ý btl s x l␴ Fllst Ž x r . s 0 s

ls1 ss1 ts1

for all x r g R r. Finally, if all the Ei jk ’s and Fl st ’s are Ž r y 1.-additi¨ e maps, then all the pi jk l st ’s are Ž r y 2.-additi¨ e and all the ␭ i jk t ’s are Ž r y 1.-additi¨ e. It is understood that if r s 1 then each pi jk t st s 0. Besides the presence of Žanti.automorphisms and derivations, it should be mentioned that there are some further improvements of earlier results on GFIs. Namely, the sum in Ž1. is no longer assumed to be equal to zero but lying in some finite dimensional space, and the maps Ei jk ’s and Fl st ’s are not assumed to be multi-additive. The applications of the theorem shall be given in the second part of this paper which is under preparation.

GENERALIZED FUNCTIONAL IDENTITIES

649

2. PRELIMINARIES We now set some further notation in place. In what follows N denotes the set of all nonnegative integers and N* s  n g N < n ) 04 . We denote by < X < the cardinality of a set X. Given q g Q, we define mappings l q , r q : Q ª Q by the rule l q Ž x . s qx,

r q Ž x . s xq for all x g Q.

Clearly l q , r q g End C Ž Q .. Setting R l s  l a < a g R4

R r s  ra < a g R4 ,

and

we note that M Ž R . s R l R r : End C Ž Q . is a subring of the ring End C Ž Q .. If h s Ý nis1 l a i r b i g M Ž R . and x g Q, then hŽ x . s Ý nis1 a i xbi . We now recall that R is GPI if and only if Q is GPI by w5, Corollary 6.1.7x. Next we state the following version of a result which is due to Erickson, Martindale, and Osborn w15, Theorem 3.1x. THEOREM 2.1 w5, Theorem 2.3.3x. Let A be a semiprime ring with extended centroid C, Q g  Q m l Ž A., Q m r Ž A.4 , n ) 1 be an integer, and let q1 , q2 , . . . , qn g Q. Suppose that q1 f Ý nis2 Cqi . Then there exists h g M Ž A. such that hŽ q1 . / 0 and hŽ qi . s 0 for all i s 2, 3, . . . , n. We also need the following result which is due to Martindale w16, Theorem 2x. THEOREM 2.2 w5, Proposition 6.3.13x. Let R be a prime ring, n g N*, and a i , bi g Q, i s 1, 2, . . . , n. Suppose that b1 , b 2 , . . . , bn are linearly independent o¨ er C Ž i.e., C-independent. and Ý nis1 a i xbi s 0 for all x g R. Then each a i s 0. As a particular case of Theorem 2.2, we note that if 0 / a, b g Q and axb s bxa for all x g R, then there exists ␭ g C with b s ␭ a Žsee also w16; 5, Theorem 2.3.4x.. Let ⌬ g W 0 , ⌬ s ␦ 1 ␦ 2 ⭈⭈⭈ ␦n where the ␦ i ’s satisfy conditions Ža. and Žb., and each ␦ i g B0 . The length < ⌬ < of ⌬ is n, the number of factors Ži.e., < ⌬ < s n.. Given a subset S s  j1 , j2 , . . . , jm 4 :  1, 2, . . . , n4 with j1 j2 - ⭈⭈⭈ - jm , we set S⬘ s  1, 2, . . . , n4 R S,

and

⌬ S s ␦ j1 ␦ j 2 ⭈⭈⭈ ␦ j m .

We call ⌬ S a subword of ⌬. Next, let ⍀ s ⌬ g g T0 , where g g G 0 . Then we call ⌬ S g a subword of ⍀. We set < ⍀ < s < ⌬ <. According to the Leibnitz formula w5, Remark 1.1.1x, ⌬ Ž xy . s Ý x ⌬ S y ⌬ S ⬘

S

for all x, y g R.

Ž 2.

650

BEIDAR, BRESAR ˇ , AND CHEBOTAR

Let u g N* and ␴ 1 , ␴ 2 , . . . , ␴u be distinct elements of T0 . We denote by L Ž ␴u . the set of all distinct subwords of ␴ 1 , ␴ 2 , . . . , ␴u . That is, L Ž ␴u . s  ␴ 1 , ␴ 2 , . . . , ␴u 4 , where u G u is the number of such subwords. In view of w5, Theorem 7.8.4x, the following result can be easily proved analogously as w4, Lemma 1x Žsee also w5, Lemma 6.1.8x.. LEMMA 2.3. Let R be a prime ring which is not GPI, let I be a nonzero ideal of R, and let Ti s  f i j Ž x . g Q² X T 0 : < j s 1, 2, . . . , n i 4 , i s 1, 2, . . . , m, be m subsets of Q² X T 0 :, each of which is C-independent. Then there exists a g I such that, for each i s 1, 2, . . . , m, Ti s  f i j Ž a. < j s 1, 2, . . . , n i 4 is a C-independent subset of Q. COROLLARY 2.4. Let R be a prime ring which is not GPI, let I be a nonzero ideal of R, let u, n g N*, n1 , n 2 , . . . , n u g N*, and let ␴ 1 , ␴ 2 , . . . , ␴n g T0 be distinct elements. Further, let  a i1 , a i2 , . . . , a i n i 4 : Q, i s 1, 2, . . . , u, let 0 / a g Q, and let W be a finite dimensional subspace of Q. Suppose that Ži. ␴ 1 s ⌬ g where ⌬ g W 0 and g g AutŽ R .. Žii. ␴ 1 f L Ž ␴i . for all i s 2, 3, . . . , n. Žiii.  a11 , a12 , . . . , a1 n 4 is C-independent. 1

Then there exists y g I such that the set

½ Ž ay .

␴1

␴1

a11 , Ž ay .

a12 , . . . , Ž ay .

␴1

a1 n1

5

is C-independent modulo the subspace ni

u

Wq

Ý Ý C Ž ay .

␴i

ai j .

is2 js1

Proof. Choose a basis  g 1Ž x ., g 2 Ž x ., . . . , g mŽ x .4 of the subspace ni

u

UsWq

Ý Ý C Ž ax .

␴i

ai j

is2 js1

of Q² X T 0 : By the Leibnitz formula Ž2., f j Ž x . s Ž ax .

␴1

a1 j s

Ý a⌬ S

S

g

x ⌬ S ⬘ g a1 j g Q² X T 0 :

651

GENERALIZED FUNCTIONAL IDENTITIES

for all j s 1, 2, . . . , n1. Note that the term in f j with exponent of the highest length is equal to a g x ␴ 1 a1 j . Now, using Žii. it follows that the set  g 1 , g 2 , . . . , g m , f 1 , f 2 , . . . , f n 4 is C-independent. The result now follows 1 from Lemma 2.3. LEMMA 2.5. Let R be a prime ring which is not GPI, let r, n i g N* where i s 1, 2, . . . , r, and let each set  qi1 , qi2 , . . . , qi n i 4 : Q, i s 1, 2, . . . , r, be C-independent. Then there exists a g R such that: Ža. and e¨ ery Žb. and e¨ ery

If Q s Q m l , then each set  aqi1 , aqi2 , . . . , aqi n i 4 is C-independent aqi j g R. If Q s Q m r , then each set  qi1 a, qi2 a, . . . , qi n i a4 is C-independent qi j a g R.

Proof. We shall prove only the first statement. Let J s  x g R < each xqi j g R 4 . By w5, Lemma 2.1.8x, J is a dense left ideal of R. Clearly, a left ideal of a prime ring with trivial right annihilator is itself a prime ring. Therefore J is a prime ring. According to w5, Proposition 2.1.10x, Q m l Ž J . s Q m l Ž R .. By Lemma 2.3 there exist a g J such that each set  qi1 a, qi2 a, . . . , qi n i a4 is C-independent. Finally we state the following important result which was first proved in w7x in the case when ␾ : R ª Q s Ž R . and in the present form appeared in w1x. LEMMA 2.6 w1, Lemma 2.13x. Let ␾ : R ª Q m l Ž R . be an additi¨ e map. Suppose that there exists an element 0 / d g R such that ␾ Ž ydx . s dy␾ Ž x . for all x, y g R. Then there exists a unique q g Q m l Ž R . such that ␾ Ž x . s dxq for all x g R. Whenever considering a sum of the form r

u

ni j

Ý Ý Ý is1 js1 ks1

g ii jk Ž x r . x i␴ j a ki j

r

or

u

ni j

Ý Ý Ý aki j x i␴ g ii jk Ž x r . j

is1 js1 ks1

we shall mean that ␴ 1 , ␴ 2 , . . . , ␴u are distinct elements of T0 , each g i jk : R ry1 ª Q and each set  a1i j, a2i j, . . . , a ni ji j 4 ; Q is C-independent. 3. THE RESULTS First we consider the case of Theorem 1.2 where only one of the two sums appears.

BEIDAR, BRESAR ˇ , AND CHEBOTAR

652

THEOREM 3.1. Let r, u g N*, r G 1, n i j g N, i s 1, 2, . . . , r, j s 1, 2, . . . , u and let V be a finite dimensional subspace of the ¨ ector space Q o¨ er C. Further, let ␴ 1 , ␴ 2 , . . . , ␴u be distinct elements of T0 , let g i jk : R ry1 ª Q, i s 1, 2, . . . , r, j s 1, . . . , u, k s 1, 2, . . . , n i j , be maps, and let each set  a1i j, a2i j, . . . , a ni ji j 4 : Q, n i j / 0, i s 1, 2, . . . , r, j s 1, 2, . . . , u, be C-independent. Suppose that R is not GPI and r

u

ni j

Ý Ý Ý

g ii jk Ž x r . x i␴ j a ki j g V

Ž 3.

is1 js1 ks1

for all x r g R r. Then each g i jk s 0. Proof. If Q s Q m r , then by Lemma 2.5 there exists b g R such that each set  a1i j b, a2i j b, . . . , a inji j b4 is C-independent and every a ikj b g R. Multiplying Ž3. by b from the right, we reduce the proof to the case when each a ikj g R. The reader will see that the case Q s Q m l Ž R . is a little bit more difficult. We now assume that Q s Q m l Ž R .. Let L Ž ␴u . s  ␴ 1 , ␴ 2 , . . . , ␴u 4 . Suppose that the theorem is false. We may assume without loss of generality that n i j / 0 implies that g i jk / 0 for all 1 F k F n i j , and Ý ris1 n i j / 0 for each j. We set r

H Ž xr . s

u

ni j

Ý Ý Ý

g ii jk Ž x r . x i␴ j a ki j

is1 js1 ks1

for all x r g R . Next, we set LŽ H . s max< ␴j < < j s 1, . . . , u4 and r

K Ž H . s Ž i , j . 1 F i F r , 1 F j F u, n i j / 0, < ␴j < s L Ž H . .

½

5

Write ␴j s ⌬ j ␣ j where ⌬ j g W 0 and ␣ j g G 0 for all j s 1, 2, . . . , u. We may assume without loss of generality that there exists ¨ with 1 F ¨ F u such that ␣ 1 , . . . , ␣ ¨ g AutŽ R . and ␣ ¨ q1 , . . . , ␣ u g AntiautŽ R .. We now make the following general observations. Fix Ž s, t . g K Ž H .. Clearly 1 F s F r, 1 F t F u, n st / 0, and < ␴t < s LŽ H .. Set ␣ s ␣ t . Consider the map ␣ : DŽ R . ª DŽ R . given by ␦ ¬ ␣y1␦␣ , ␦ g DŽ R .. According to w5, p. 301x, ␣ is a semilinear differential C-Lie algebra automorphism of DŽ R . which can be uniquely extended to a semilinear automorphism ␣ ˆ of the universal enveloping algebra U of the Žrestricted. differential C-algebra DŽ R .. By w5, Lemma 7.2.1x, B0␣ is a right C-basis for DŽ R . modulo Di and ␣y1 G 0 is a set of representatives of y1 G modulo Gi . Set T0X s W 0␣ˆ␣y1 G 0 . Substituting x i␣ for x i , i s 1, 2, . . . , r, in Ž3., we get r

u

ni j

Ý Ý Ý is1 js1 ks1

g ii jk Ž x r . x i␴ j a ikj g V ,

653

GENERALIZED FUNCTIONAL IDENTITIES

where g ii jk Ž x r . s g ii jk Ž x r␣

y1

and

.

␣ ˆ

␴j s Ž ⌬ j . ␣y1␣ j .

In particular ␴t g W 0␣ˆ. Therefore we may assume without loss of generality that ␴t g W 0 . Further, by Lemma 2.5 there exists b g R such that ba1st , ba2st , . . . , bansts t are C-independent and each bakst g R. Since ␴t g W 0 , the Leibnitz formula Ž2. implies that HˆŽ x r . s H Ž x 1 , . . . , x sy 1 , x s b, x sq1 , . . . , x r . can be written as Hˆ Ž x r . s

n st

Ý

mi j

g sts k Ž x r . x s␴ t bakst q

ks1

Ý

Ý ˆg ii jk Ž x r . x i␴ bki j g V . Ž 4. j

1FiFr , 1FjFu , ks1 Ž i , j ./ Ž s, t .

Clearly LŽ Hˆ . s LŽ H . and K Ž Hˆ . : K Ž H .. Since our goal is to prove that each g st k must be equal to 0 Žand thus obtain a contradiction., we may assume without loss of generality that for any fixed Ž s, t . g K Ž H . the following conditions are satisfied: ␴t g W 0 and each a kst g R. Recall that N is the set of nonnegative integers. We order N = N lexicographically. Clearly the set N = N is well-ordered with respect to this order which we denote by - . We proceed by induction on Ž LŽ H ., < K Ž H .<. Žhere, < K Ž H .< means the cardinality of K Ž H ... First assume that Ž LŽ H ., < K Ž H .<. s Ž0, 1.. Since LŽ H . s 0, each ␴j g G 0 . As < K Ž H .< s 1, we may assume without loss of generality that n s n11 / 0 and n i j s 0 for all Ž i, j . / Ž1, 1.. By the above observations we may assume without loss of generality that ␴ 1 s 1 and each a11 k g R. Fix x 2 , x 3 , . . . , x r g R such that g 111Ž x 2 , . . . , x r . / 0 and set a k s g 11 k Ž x 2 , . . . , x r . and bk s a11 k for all k s 1, 2, . . . , n. Choose a g R such that aa1 / 0 and each aak g R. Multiplying Ž3. by a from the left, we get n

Ý aak x 1 bk g aV

for all x 1 g R.

ks1

As R is not GPI, w16, Theorem 2x yields that each aak s 0, a contradiction. Therefore the theorem is true if Ž LŽ H ., < K Ž H .<. s Ž0, 1.. In the inductive case we may assume without loss of generality that Ž r, 1. g K Ž H ., ␴ 1 g W 0 and each a kr1 g R. Now assume that n r1 s 1. Fix y g R, and set a s a1r1 and F Ž x r . s H Ž x 1 , x 2 , . . . , x r ay . y H Ž x r . ya. Clearly F Ž x r . g V q Vya

for all x r g R r

BEIDAR, BRESAR ˇ , AND CHEBOTAR

654

and LŽ F . F LŽ H .. If LŽ F . s LŽ H ., then K Ž F . : K Ž H .. Since Ž r, 1. f K Ž F ., < K Ž F .< - < K Ž H .<. Therefore Ž LŽ F ., < K Ž F .<. - Ž LŽ H ., < K Ž H .<. and we can apply the induction assumption. We now specify the choice of y. Since Ž LŽ H ., < K Ž H .<. ) Ž0, 1., there are only the following four cases. Case 1. Suppose that LŽ H . s 0 and n i j / 0 for some 1 F i - r and 1 F j F u. Then by Lemma 2.3 there exists y g R such that the set ij 1 ,

½a

a2i j , . . . , a inji j , a1i j ya, a2i j ya, . . . , a inji j ya

5

is C-independent. Applying the induction assumption to F one easily gets that in particular each g i jk s 0, a contradiction. Case 2. Assume that LŽ H . s 0 and n r j / 0 for some 2 F j F ¨ . Recall that 1 / ␴j g AutŽ R .. By Lemma 2.3 there exists y g R such that the set

½ Ž ay .

␴j

a1r j , Ž ay .

␴j

a2r j , . . . , Ž ay .

␴j

a nr jr j , a1r j ya, a2r j ya, . . . , a nr jr j ya

5

is C-independent. Again the induction assumption yields in particular that each g r jk s 0, a contradiction. Case 3. Suppose that LŽ H . s 0 and n r j / 0 for some ¨ q 1 F j F u. Recall that 1 / ␴j g AntiautŽ R .. It follows from Lemma 2.3 that there exists y g R such that the set rj 1 ,

½a

a2r j , . . . , a nr jr j , a1r j ya, a2r j ya, . . . , a nr jr j ya

5

is C-independent. Again the induction assumption yields in particular that each g r jk s 0, a contradiction. Case 4. Finally assume that LŽ H . ) 0. We set w s j if ␴j s 1 for some 1 F j F ¨ ; otherwise we set w s 0 and n r w s 0. By Corollary 2.4 there exists y g R such that nr i

␴

Ž ay . 1 a f

Ý Ý C Ž ay .

␴i

nr w

a rj i q

2FiFu , js1 ␴ ig W 0

Ý Ca rj w ya.

Ž 5.

js1

The Leibnitz formula Ž2. implies that F can be written in the form r

F Ž xr . s

mi j

u

Ý Ý Ý h ii jk Ž x r . x i␴ bki j g V q Vya j

for all x r g R r . Ž 6 .

is1 js1 ks1

Clearly ␴p s 1 for some 1 F p F u. We may also assume without loss of generality that b1r p s Ž ay . ␴1 a and nr i

bkr p g

Ý Ý C Ž ay . 2FiFu , js1 ␴ ig W 0

␴i

nr w

a rj i q

Ý Ca rj w ya, js1

k s 2, 3, . . . , m r p .

GENERALIZED FUNCTIONAL IDENTITIES

655

Computing left and right coefficients of x r in Ž6. and using Ž5., one gets that h r p1 s g r11. By the induction assumption Ž6. yields in particular that g r11 s 0, a contradiction. Finally, assume that n r1 ) 1 and let us show again that g r11 s 0. By Ž . Ž r1 . Theorem 2.1 there exits h s Ý m ts1 l b t r c t g M R such that h a1 s a / 0 r1 and hŽ a k . s 0 for all k s 2, 3, . . . , n r1 , where all the bt ’s and c t ’s are in R. Set m

F Ž xr . s

Ý H Ž x 1 , x 2 , . . . , x r bt . c t . ts1

r Ž . Clearly F Ž x r . g Ý m ts1Vc t for all x r g R . The Leibnitz formula 2 implies that F can be written in the form

mi j

Fs

Ý

Ý

f iijk Ž x 1 , . . . , x r . x i␴ j d ki j q g r11 Ž x ry1 . x r␴ 1 a.

1FiFr , 1FjFu , ks1 Ž i , j ./ Ž r , 1 .

Clearly K Ž F . : K Ž H . and so < K Ž F .< F < K Ž H .<. By the above result g r11 s 0, a contradiction. The proof is now complete. Analogously one can prove the following result. THEOREM 3.2. Let r, u g N*, r G 1, n i j g N, i s 1, 2, . . . , r, j s 1, 2, . . . , u and let V be a finite dimensional subspace of the ¨ ector space Q o¨ er C. Further, let ␴ 1 , ␴ 2 , . . . , ␴u be distinct elements of T0 , let g i jk : R ry1 ª Q, i s 1, 2, . . . , r, j s 1, . . . , u, k s 1, 2, . . . , n i j , be maps, and let each set  a1i j, a2i j, . . . , a ni ji j 4 : Q, n i j / 0, i s 1, 2, . . . , r, j s 1, 2, . . . , u, be C-independent. Suppose that R is not GPI and r

u

ni j

Ý Ý Ý aikj x i␴ g ii jk Ž x r . g V j

is1 js1 ks1

for all x r g R r. Then each g i jk s 0. COROLLARY 3.3. Let r, u g N*, r G 1, and n j g N, j s 1, 2, . . . , u. Let ␴ 1 , . . . , ␴u be distinct elements of T0 , let Ejk , F: R ry1 ª Q, j s 1, 2, . . . , u, k s 1, 2, . . . , n j , be maps, and let  a1j , a2j , . . . , a nj j 4 be a C-independent subset of Q for all j s 1, 2, . . . , u for which n j / 0. Next, let 0 / b g R and let V be a finite dimensional subspace of Q. Suppose that R is not GPI, ␴u s 1, and u

nj

Ý Ý Ejk Ž x ry1 . x r␴ akj q bx r F Ž x ry1 . g V j

js1 ks1

for all x r g R r .

Ž 7.

BEIDAR, BRESAR ˇ , AND CHEBOTAR

656 Then:

Ejk s 0 for all j s 1, 2, . . . , u y 1, k s 1, 2, . . . , n j . There exist maps ␭ k : R ry1 ª C such that each Eu k Ž x ry1 . s 1 Ž . Ž . u ␭ k x ry1 b and F Ž x ry1 . s yÝ ny ks1 ␭ k x ry1 a k . Ž1. Ž2.

Proof. Let L Ž ␴u . s  ␴ 1 , ␴ 2 , . . . , ␴u4 . We may assume without loss of generality that each Ejk / 0. Set m s max< ␴j < < j s 1, 2, . . . , u4 . Write ␴i s ⌬ i ␣ i where ⌬ i g W 0 and ␣ i g G 0 . Set nj

u

H Ž xr . s

Ý Ý Ejk Ž x ry1 . x r␴ akj q bx r F Ž x ry1 . . j

js1 ks1

First we claim that each ␣ j g AutŽ R .. If not, then without loss of generality we may assume ␣ 1 g AntiautŽ R . and < ␴ 1 < is maximal possible among all the ␴t with ␣ t g AntiautŽ R .. Clearly ␴ 1 is not a subword of any ␴j with j ) 1. By Lemma 2.3 there exists y g R such that the set

 b ␣ y ␣ a11 , b ␣ y ␣ a12 , . . . , b ␣ y ␣ a1n , a11 , a12 , . . . , a1n 4 , 1

1

1

1

1

1

1

1

is C-independent. Computing H Ž x ry1 , ybx r . y byH Ž x r ., according to the Leibnitz formula Ž2. we obtain n1

Ý E1 k Ž x ry1 . x r␴ b ␣ 1

1

y ␣ 1 a1k y

ks1

n1

Ý byE1 k Ž x ry1 . x r␴

1

a1k

ks1 nj

u

q

Ý Ý Ejk Ž x ry1 . x r␴ akj g V q byV j

js2 ks1

for all x r g R r. By Theorem 3.1 we have in particular that E1 k Ž x ry1 . s 0 for all k s 1, 2, . . . , n1. Since by assumption each Ejk / 0, we conclude that each ␣ j g AutŽ R .. Next we claim that u s 1. Assume that u ) 1. Fix any t with 1 F t F u and set J s  j < 1 F j F u, ␣ j s ␣ t 4 . Clearly t g J. Computing H Ž x ry1 , zbx r . y bzH Ž x r ., as above we obtain u

nj

Ý Ý Ejk Ž x ry1 . x r␴ akj g V q bzV j

js1 ks1

GENERALIZED FUNCTIONAL IDENTITIES

657

for all x r g R r. Obviously ␴ l s ␣ t for some 1 F l F u. Clearly each map Eu k is a C-linear combination of maps

Gjs

¡E w zb x s~ ¢E w zb x

␴j

if j g J , < ␴j < ) 0, 1 F s F n j ,

js

␴j

js

if j g J , < ␴j < s 0, 1 F s F n j

y bzEjs

Žhere it is understood that Ejs s Ejs Ž x ry1 ., etc... By Theorem 3.1 we have in particular that El1 s 0. Fixing x ry1 g R ry1 such that at least one Ejs Ž x ry1 . involved in El1 is nonzero, we obtain a nonzero reduced T ⬘identity in z, which is impossible. Therefore u s 1 and so n

H Ž xr . s

Ý Ek Ž x ry1 . x r ak q bx r F Ž x ry1 . , ks1

where n s n1 , Ek s E1 k , and a k s a1k for all k s 1, 2, . . . , n. Again computing H Ž x ry1 , zbx r . y bzH Ž x r . and applying Theorem 3.1, we obtain that each Ek Ž x ry1 . zb y bzEk Ž x ry1 . s 0 for all z g R. It follows from Theorem 2.2 that Ek Ž x ry1 . g Cb for all x ry1 g R ry1. Therefore there exist maps ␭ k : R ry1 ª C such that each Ek Ž x ry1 . s ␭ k Ž x ry1 . b. It now follows that n

bx r F Ž x ry1 . q

Ý ␭k Ž x ry1 . ak

gV

ks1

for all x r g R r. The result now follows from Theorem 3.2. LEMMA 3.4. Let r, u g N*, r G 1, n i j , m g N, i s 1, 2, . . . , r, j s 1, 2, . . . , u. Further, let ␴ 1 , ␴ 2 , . . . , ␴u be distinct elements of T0 , let g i jk : R ry1 ª Q, ␮ l : R r ª C, i s 1, 2, . . . , r, j s 1, . . . , u, k s 1, 2, . . . , n i j , l s 1, 2, . . . , m, be maps, let each set  a1i j, a2i j, . . . , a ni ji j 4 : Q, n i j / 0, i s 1, 2, . . . , r, j s 1, 2, . . . , u, be C-independent, and let  b1 , b 2 , . . . , bm 4 , m / 0 be a C-independent set. Suppose that R is not GPI and r

H Ž xr . s

u

ni j

Ý Ý Ý is1 js1 ks1

g ii jk Ž x r . x i␴ j a ikj q

m

Ý ␮ l Ž x r . bl ls1

is an r-additi¨ e map R r ª Q. Then each g i jk is Ž r y 1.-additi¨ e and each ␮ l is r-additi¨ e.

BEIDAR, BRESAR ˇ , AND CHEBOTAR

658

Proof. It is enough to show that each ␮ l and each g i jk with i - r are additive in x r . To this end fix xXr , xYr g R. Since H Ž x ry1 , xXr q xYr . y H Ž x ry1 , xXr . y H Ž x ry1 , xYr . s 0, we conclude that ry1

ni j

u

Ý Ý Ý

g i jk Ž x ry1 , xXr q xYr . y g i jk Ž x ry1 , xXr . y g i jk Ž x ry1 , xYr . x i␴ j a ki j

is1 js1 ks1 m

s

Ý

␮ l Ž x ry1 , xXr . q ␮ l Ž x ry1 , xYr . y ␮ l Ž x ry1 , xXr q xYr . bl

ls1 m

g

Ý Cbl ls1

for all x ry1 g R ry1. By Theorem 3.1 each g i jk Ž x ry1 , xXr q xYr . y g i jk Ž x ry1 , xXr . y g i jk Ž x ry1 , xYr . s 0. Since  b1 , b 2 , . . . , bm 4 is a C-independent set, we get that each

␮ l Ž x ry1 , xXr . q ␮ l Ž x ry1 , xYr . y ␮ l Ž x ry1 , xXr q xYr . s 0. The proof is complete. Proof of Theorem 1.2. Without loss of generality we may assume that Q s Q m l Ž R .. Consider Ž1.. Write ␴j s ⌬ j ␣ j where each ⌬ j g W 0 and each ␣ j g G 0 . Let L Ž ␴u . s  ␴ 1 , ␴ 2 , . . . , ␴u 4 and let 1 F l F r, 1 F s F u and 1 F t F m l s . We shall say that the map Fl st has a standard presentation if u

Fllst

ni j

nl s

Ž x r . s y Ý Ý Ý pii jkl l st Ž x r . x i␴ aikj y Ý ␭ll sk t Ž x r . alks , j

1FiFr js1 ks1 i/l

ks1

where pi jk l st : R ry2 ª Q, ␭ l sk t : R ry1 ª C, and the ␴j ’s, a ikj ’s are from Ž1.. Next, let S :  Ž l, s, t . 1 F l F r , 1 F s F u, 1 F t F m l s 4 . We shall say that Fl st with Ž l, s, t . g S has a quasi-standard presentation if there exist n i j g N, a ikj g Q m l , pi jk l st : R ry2 ª Q m l Ž R . and ␭ l sk t : R ry1 ª C

659

GENERALIZED FUNCTIONAL IDENTITIES

such that ni j

u

Fllst

Ž x r . s y Ý Ý Ý piiljk l st Ž x r . x i␴ aikj j

1FiFr js1 ks1 i/l nl s

y

Ý ␭ll sk t Ž x r . akl s ,

Ž l, s, t . g S,

ks1

where Žc. n i j G n i j for all i s 1, 2, . . . , r, j s 1, 2, . . . , u. Žd. Each set  a ikj < k s 1, 2, . . . , n i j 4 is C-independent. Že. a ikj s a ikj for all i s 1, 2, . . . , r, j s 1, 2, . . . , u, k s 1, 2, . . . , n i j . Clearly a standard presentation of Fl st is also a quasi-standard presentation of it. Further, we set r

H Ž xr . s

u

ni j

Ý Ý Ý

Eii jk Ž x r . x i␴ j a ikj q

is1 js1 ks1

r

u

ml s

Ý Ý Ý btl s x l␴ Fllst Ž x r . , s

ls1 ss1 ts1

½

L Ž H . s max < ␴s < 1 F s F u,

r

Ý ml s / 0 ls1

5

,

K Ž H . s  Ž l, s . 1 F l F r , 1 F s F u, < ␴s < s L Ž H . , m l s / 0 4 . As in the proof of Theorem 3.1, we proceed by induction on Ž LŽ H ., < K Ž H .<.. First we make the following general observations. ŽA. Each Ei jk has a standard presentation if and only if each Fl st has a standard presentation. Indeed, let, say, each Fl st have a standard presentation. Then Ž1. implies that r

u

ni j

Ý Ý Ý is1 js1 ks1

u

Eii jk Ž x r . y

mls

Ý Ý Ý btl s x l␴ piiljk l st Ž x r . s

1FlFr ss1 ts1 l/i mi j

y

Ý ␭ii jk t Ž x r . bti j

x i␴ j a ki j g V

ts1

for all x r g R r. Theorem 3.1 now yields that each Ei jk has a standard presentation. ŽB. Each Fl st has a standard presentation if and only if each Fl st has a quasi-standard presentation. Indeed, let all the Fl st ’s have quasi-standard

BEIDAR, BRESAR ˇ , AND CHEBOTAR

660

presentations. Set Ei jk s 0 if either j ) u or k ) n i j . Then Ž1. yields that r

u

ni j

Ý Ý Ý

mls

u

Ý Ý Ý btl s x l␴ piiljk l st Ž x r .

Eii jk Ž x r . y

is1 js1 ks1

s

1FlFr ss1 ts1 l/i mi j

x i␴ j a ki j g V

Ý ␭ii jk t Ž x r . bti j

y

ts1

for all x r g R r. By Theorem 3.1 each Ei jk has a standard presentation. The result now follows from ŽA.. ŽC. We may assume without loss of generality that each btl s g R. Indeed, by Lemma 2.5 there exists a g R such that each set  ab1l s, ab2l s, . . . , abml s 4 is C-independent and each abtl s g R. Multiplying Ž1. ls by a from the left we see that r

aH Ž x r . s

u

ni j

r

u

ml s

Ý Ý Ý aEii jk Ž x r . x i␴ aikj q Ý Ý Ý abtl s x l␴ Fllst Ž x r . g aV j

s

is1 js1 ks1

ls1 ss1 ts1

for all x r g R r. Clearly LŽ aH . s LŽ H . and K Ž aH . s K Ž H .. If the theorem is true for aH, then each Fl st is of the standard form and so each Ei jk is of the standard form by ŽA. which proves our statement. In what follows we shall assume that each btl s g R. ŽD. Fix Ž l, s . g K Ž H .. Arguing as in the proof of Theorem 3.1, we reduce the proof to the case when ␴s g W 0 , i.e., ␣ s s 1. Further, it is enough to show that all the Fl st ’s have quasi-standard presentations. Indeed, let all the Fl st ’s have quasi-standard presentations. For simplicity we may assume that Ž l, s . s Ž1, 1.. We have r 1 F11 t

u

ni j

Ž xr . s y Ý Ý Ý

pii1jk11 t

Ž xr .

x i␴ j a ikj

is2 js1 ks1

n11

y

Ý ␭111 k t Ž x r . a11k

Ž 8.

ks1

for all t s 1, 2, . . . , m11 . Set Ei jk s 0 if either j ) u or k ) n i j . Next, set m l s s 0 if s ) u. According to Že. we can now rewrite Ž1. as r

u

ni j

r

u

mls

Ý Ý Ý Eii jk Ž x r . x i␴ aikj q Ý Ý Ý btl s x l␴ Fllst Ž x r . g V . j

is1 js1 ks1

s

ls1 ss1 ts1

Ž 9.

661

GENERALIZED FUNCTIONAL IDENTITIES

Further, we set m11

Eii jk Ž x r . s Eii jk Ž x r . y

Ý bt11 x 1␴

1

pii1jk11 t Ž x r . ,

ts1

2 F i F r , 1 F j F u, 1 F k F n i j , E11 jk Ž x r . s E11 jk Ž x r . ,

2 F j F u, 1 F k F n i j ,

m11 1 1 E11 k Ž x r . s E11 k Ž x r . y

Ý ␭111 k t Ž x r . bt11 ,

1 F k F ni j .

ts1

Substituting Ž8. in Ž9. we obtain that r

H Ž xr . s

ni j

u

Ý Ý Ý Eii jk Ž x r . x i␴ aikj j

is1 js1 ks1 ml s

q

Ý

Ý btl s x l␴ Fllst Ž x r . g V s

1FlFr , 1FsFu , ts1 Ž l , s ./ Ž1, 1 .

for all x r g R r. If < K Ž H .< ) 1, then LŽ H . s LŽ H . and K Ž H . : K Ž H .. Since Ž1, 1. f K Ž H ., < K Ž H .< - < K Ž H .< and so the induction assumption implies that all the Fl st ’s have quasi-standard presentations. If < K Ž H .< s 1 Ži.e., K Ž H . s Ž1, 1.4., then LŽ H . - LŽ H . and again the induction assumption implies that all the Fl st ’s have quasi-standard presentations. The result now follows from ŽB.. ŽE. We may assume without loss of generality that Ž r, 1. g K Ž H .. By ŽD. we may assume that ␴ 1 g W 0 and ␣ 1 s 1. We also may assume without loss of generality that m r1 s 1. Indeed, suppose that m r1 ) 1. In order to prove the theorem, it is enough to show that all the Fr1 t ’s have quasi-standard presentations. To this end it is enough to show that Fr1 m r1 has a quasi-standard presentation because then we can decrease m r1 by 1 via replacement of Fr1 m r1 by its quasi-standard presentation and proceed by induction on m r1. By Theorem 2.1 there exist elements c p , d p g R such that Ý p c p btr1 d p s 0 for all t - m r1 and b s Ý p c p bmr1r 1 d p / 0. Set H Ž x r . s Ý p c p H Ž x ry1 , d p x r . and note that H Ž x r . g Ý p c pV for all x r g R r. It follows from the Leibnitz formula Ž2. that H can be written in the form r

H Ž xr . s

u

ni j

Ý Ý Ý

Eii jk

Ž xr .

is1 js1 ks1 u

q

x i␴ j a ikj

ry1

q

u

mls

Ý Ý Ý btl s x l␴ Fllst Ž x r . s

ls1 ss1 ts1

mrs

Ý Ý btr s x r␴ Fr st Ž x ry1 . q bx r␴ Fr1 m Ž x ry1 . , s

1

r1

ss2 ts1

BEIDAR, BRESAR ˇ , AND CHEBOTAR

662

where for all s s u q 1, u q 2, . . . , u either < ␴s < - LŽ H . or m r s s 0, because each ␴s with s ) u and m r s / 0 is a proper subword of some ␴j with j F u Ži.e., it is a subword of ␴j and ␴s / ␴j . and m r j / 0 and so < ␴s < - < ␴j < F LŽ H .. Further we may assume without loss of generality that the n i j ’s and a ki j ’s are chosen in such a way that the conditions Žc. ᎐ Žd. are fulfilled Žwe just extend each set  a1i j, a2i j, . . . , a ni ji j 4 to a basis of the vector subspace of Q generated by all the right coefficients of x i␴ j appearing in the first sum in the presentation of H .. If the theorem is true for H, then Fr1 m r1 has a quasi-standard presentation. The proof of our claim is complete. ŽF. If < K Ž H .< s 0, the result follows from Theorem 3.1. Therefore we may assume without loss of generality that < K Ž H .< / 0. Now suppose that Ž LŽ H ., < K Ž H .<. s Ž0, 1.. We may asume without loss of generality that K Ž H . s Ž r, 1.4 , ␴ 1 s 1, and m r1 s 1 Žsee ŽD. and ŽE... Set b s b1r1. Now Ž1. reads r

ni j

u

Ý Ý Ý Eii jk Ž x r . x i␴ aikj q bx r Fr11Ž x ry1 . g V j

Ž 10 .

is1 js1 ks1

for all x r g R r. Fix x ry1 g R. Since bx r Fr1 t Ž x ry1 . is additive in x r , Lemma 3.4 yields that each Ei jk , i - r, is additive in x r . Let y g R and set H Ž x r . s H Ž x ry1 , ybx r . y byH Ž x r .. Clearly H Ž x r . g V q byV for all x r g R r and ry1

H Ž xr . s

u

ni j

Ý Ý Ý

Eii jk Ž x ry1 , ybx r . y byEii jk Ž x r . x i␴ j a ki j

is1 js1 ks1 u

q

nr j

Ý Ý Er jk Ž x ry1 . x r␴ akr j . j

js1 ks1

Since H Ž x r . g V q byV for all x r g R r , Theorem 3.1 implies that in particular Eii jk Ž x ry1 , ybx r . s byEii jk Ž x r . for all i, j, k with i - r. It follows from Lemma 2.6 that for each x ry1 g R ry1 and i, j, k with i - r there exists a uniquely determined element pii jk r11Ž x ry1 . g Q m l Ž R . such that Eii jk Ž x r . s bx r pii jk r11 Ž x ry1 . s b1r1 x r␴ 1 pii jk r11 Ž x ry1 . .

Ž 11 .

663

GENERALIZED FUNCTIONAL IDENTITIES

Substituting Ž11. into Ž1. we obtain nr j

u

H Ž xr . s

Ý Ý Er jk Ž x ry1 . x r␴ akr j q bx r F Ž x ry1 . g V

for all x r g R r ,

j

jsr ks1

where ry1

F Ž x ry1 . s Fr11 Ž x ry1 . q

u

ni j

Ý Ý Ý

pii jk r11 Ž x ry1 . x i␴ j a ki j .

is1 js1 ks1

The result now follows from Corollary 3.3. In the inductive case Ž LŽ H ., < K Ž H .<. ) Ž0, 1. we may also assume without loss of generality that Ž r, 1. g K Ž H ., ␴ 1 g W 0 and m r1 s 1. By ŽD. it is enough to show that Fl s1 has a quasi-standard presentation for some 1 F l F r and 1 F s F u with Ž l, s . g K Ž H .. Let y g R. Set b s b1r1 and H Ž x r . s H Ž x ry1 , ybx r . y byH Ž x r .. Clearly H Ž x r . g V q byV for all x r g R r and it follows from the Leibnitz formula Ž2. that H can be written as r

H Ž xr . s

u

ni j

Ý Ý Ý

Eii jk Ž x r . x i␴ j a ikj q

is1 js1 ks1

mls

Ý

Ý btl s x l␴ Fllst Ž x r . . s

1FlFr , 1FsFu , ts1 Ž l , s ./ Ž r , 1 .

Ž 12 . Clearly LŽ H . F LŽ H .. We may assume without loss of generality that n i j G n i j and the a ikj ’s are chosen in such a way that a ki j s a ki j for all i s 1, 2, . . . , r, j s 1, 2, . . . , u, k s 1, 2, . . . , n i j . If LŽ H . s LŽ H ., then K Ž H . : K Ž H .. As Ž r, 1. f K Ž H ., we conclude that < K Ž H .< - < K Ž H .<. Therefore Ž LŽ H ., < K Ž H .<. - Ž LŽ H ., < K Ž H .<. and so we can apply the induction assumption to H. We now specify the choice of y. Since Ž LŽ H ., < K Ž H .<. ) Ž0, 1., there are only the following four cases. Case 1. Suppose that LŽ H . s 0 and m p q / 0 for some 1 F p - r and 1 F q F u. Then u s u, Ž p, q . g K Ž H . and it is enough to show that all the Fp q t ’s have quasi-standard presentations. By Lemma 2.3 there exists y g R such that the set

 b1p q , b2p q , . . . , bmp q , byb1p q , byb2p q , . . . , bybmp q 4 pq

pq

is C-independent and so we may assume without loss of generality that m p q G m p q , btp q s bybtp q and Fp q t s Fp q t for all t s 1, 2, . . . , m p q . Applying the induction assumption to H one easily gets that, in particular, each Fp q t has a quasi-standard presentation.

BEIDAR, BRESAR ˇ , AND CHEBOTAR

664

Case 2. Assume that LŽ H . s 0 and m r q / 0 for some 2 F q F u with ␴q g AutŽ R .. Obviously Ž r, q . g K Ž H . and so it is enough to show that each Fr q t has a quasi-standard presentation. By Lemma 2.3 there exists y g R such that the set

½b

rq 1

␴

␴

␴

Ž yb . q , b 2r q Ž yb . q , . . . , bmr qr qŽ yb . q , byb1r q , byb2r q , . . . , bybmr qr q

5

is C-independent. Again the induction assumption yields in particular that each Fr q t has a quasi-standard presentation. Case 3. Suppose that LŽ H . s 0 and m r q / 0 for some 1 F q F u with ␴q g AntiautŽ R .. We see that Ž r, q . g K Ž H . and as above it is enough to show that each Fr q t has a quasi-standard presentation. By Lemma 2.3 there exists y g R such that the set

 b1r q , b2r q , . . . , bmr q , byb1r q , byb2r q , . . . , bybmr q 4 rq

rq

is C-independent. Again the induction assumption yields in particular that each Fr q t has a quasi-standard presentation. Case 4. Finally assume that LŽ H . ) 0. We set w s j if ␴j s 1 for some 1 F j F u; otherwise we set w s 0 and m r w s 0. By Corollary 2.4 there exists y g R such that b Ž yb .

␴1

mr s

f

Ý

Ý Cbtr s Ž yb .

␴s

mr w

q

2FsFu , ts1 ␴sg W 0

Ý Cbybtr w

Ž 13 .

ts1

because the condition < ␴ 1 < s LŽ H . implies that ␴ 1 is not a subword of any ␴s with s ) 1. Turning back our attention to Ž12., in view of Ž13. we may assume without loss of generality that the bkr u ’s are chosen in such a way that b1r u s bŽ yb . ␴ 1 and mrs

bqr u g

Ý

Ý

2FsFu , ps1 ␴sg W 0

Cbpr s Ž yb .

␴s

mr w

q

Ý Cbybtr w ,

q s 2, 3, . . . , m r1 .

ts1

It now follows from Ž12. that Fr11 s Fr11. Therefore the induction assumption implies that in particular Fr11 s Fr11 has a quasi-standard presentation. The proof is now complete.

ACKNOWLEDGMENT The authors thank the referee for the careful reading of the paper and for several useful suggestions.

GENERALIZED FUNCTIONAL IDENTITIES

665

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