Generalized Derivations with Power Central Values on Multilinear Polynomials on Right Ideals

Generalized Derivations with Power Central Values on Multilinear Polynomials on Right Ideals.

N. ARGACË(*) - V. DEFILIPPIS(**) - H.G. INCEBOZ(*)

ABSTRACT- LetK be a commutative ring with unity,Ra prime K-algebra, with extended centroidCand right Utumi quotient ringU,ga non-zero generalized derivation ofR. Suppose thatf(x1;. . .:;xn) is a multilinear polynomial overK,I is a non-zero right ideal ofRandm1, a fixed integer. We prove the following results:

If g(f(r1;. . .:;rn))^mˆ0, for all r1;. . .:;rn2I, then one of the following holds:

(1) [f(x1;. . .:;xn);xn‡1]xn‡2is an identity forI;

(2) g(x)ˆaxfor allx2R, wherea2Usuch thataIˆ0;

(3) g(x)ˆax‡[q;x] for allx2R, wherea;q2Usuch thataIˆ0 and [q;I]Iˆ0.

If there exist a1;. . .;an2I such that g(f(a1;. . .:;an))^m 6ˆ0 and g(f(r1;. . .;rn))^m2Z(R), for allr1;. . .:;rn2I, then one of the following holds:

(1) f(x1;. . .;xn)xn‡1is an identity forI;

(2) f(x1;. . .;xn) is central valued onR;

(3) g(x)ˆaxfora2Candf(x1;. . .;xn) is power central valued onR;

(4) Rsatisfiess4, the standard identity in four variables.

1. Introduction.

Throughout this paper,Ris always a prime ring with centerZ(R), ex- tended centroidC, and right Utumi quotient ringU. An additive mapping d:R!Ris called derivation ifd(xy)ˆd(x)y‡xd(y) holds for allx;y2R.

The study of derivations of prime rings was initiated by Posner [20]. By a

(*) Indirizzo degli AA.: Department of Mathematics, Ege University, Science Faculty, 35100, Bornova, Izmir, Turkey.

E-mail: [email protected]

(**) Indirizzo dell'A.: Faculty of Engineering, University of Messina, Contrada Di Dio, 98166, Messina, Italy

E-mail: [email protected]

generalized derivation onRone usually means an additive mapg:R!R such that, for anyx;y2R,g(xy)ˆg(x)y‡xd(y), for some derivationdin R. Obviously any derivation is a generalized derivation. Moreover, other basic examples of generalized derivations are the following: (i) g(x)ˆ

ˆax‡xb, fora;b2R; (ii)g(x)ˆax, for somea2R:Many authors have studied generalized derivations in the context of prime and semiprime rings (see [13], [10], [17]).

In [7], Giambruno and Herstein proved that ifdis a derivation ofRsuch thatd(x)ⁿˆ0 for allx2R, thendˆ0. Following this line of investigation in [8] Herstein showed that ifd(x)ⁿ2Z(R), for allx2R, then eitherdˆ0 orR satisfies the standard identitys₄. Later in [23], Wong considered a similar condition in case the derivation acts on a multilinear polynomial f(x1;. . .;xn). The conclusion was that eitherf(x1;. . .;xn) is central inRor Rsatisfiess₄. Recently in [2], Chang extended Wong's result to the case when thead(f(x₁;. . .;x_n))^mˆ0, for allx₁;. . .;x_nin a non-zero right idealI ofR,a2Randmis a fixed integer. He concluded that either aIˆ0 or d(I)Iˆ0 or [f(x1;. . .;xn);xn‡1]xn‡2is an identity forI. At this point the natural question is what happens in case the derivation is replaced by a generalized derivation. More recently in [[22], Theorem 5], Wei proved that ifRis a prime ring,Ia non-zero ideal ofR,ma fixed positive integer and gis a generalized derivation such that g(x)^mˆ0, for allx2I, then gˆ0. Finally in a very recent paper [21] Wang generalized above theorem to the case when the generalized derivation acts on all evaluations of a multilinear polynomial in a prime ring. More precisely he proved that ifgis a generalized derivation of a prime K-algebra R and f(x₁;. . .;x_n) is a multilinear polynomial over K, such that g(f(r₁;. . .;r_n))^mˆ0 for all

r1;. . .;rn2R, then either there exists an element l2C such that

g(x)ˆlxfor allx2R, orf(x1;. . .;xn) is central valued onR, except when Rsatisfiess4 the standard identity in four variables.

In this paper we will continue this line of investigation. Our aim is to extend the Wang's result to the case when the multilinear polynomial is evalued on one-sided ideals of the primeK-algebraR. We will prove that:

THEOREM1. Let K be a commutative ring with unity, R a prime K- algebra, with extended centroid C, g a non-zero generalized derivation of R,

f(x₁;. . .:;x_n)a multilinear polynomial over K, I a non-zero right ideal of R

and m1, a fixed integer. If g(f(r₁;. . .:;r_n))^mˆ0, for all r₁;. . .:;r_n2I, then one of the following holds:

(i) [f(x₁;. . .:;x_n);x_n‡1]x_n‡2 is an identity for I;

(ii) g(x)ˆax for all x2R, where a2U such that aIˆ0;

(iii) g(x)ˆax‡[q;x]for all x2R, where a;q2U such that aIˆ0 and[q;I]Iˆ0.

THEOREM2. Let K be a commutative ring with unity, R a prime K- algebra, with extended centroid C, g a non-zero generalized derivation of R. Suppose that f(x1;. . .:;xn)is a multilinear polynomial over K, I is a non-zero right idealof R and m1, a fixed integer such that g(f(r1;. . .:;rn))^m2Z(R), for al l r1;. . .;rn2I. If there exist a1;. . .;an2I such that g(f(a₁;. . .:;a_n))^m6ˆ0, then one of the following holds:

(i) f(x1;. . .;xn)xn‡1 is an identity for I;

(ii) f(x₁;. . .;xn)is centralvalued on R;

(iii) g(x)ˆax for a2C and f(x₁;. . .;x_n)is power centralvalued on R;

(iv) R satisfies s₄, the standard identity in four variables.

2. Preliminaries.

In all that follows, unless stated otherwise,Kwill be a commutative ring with unity andRwill be a primeK-algebra. The related object we need to mention is the right Utumi quotientUof a ringR(sometimes, as in [1],U is called the maximal right ring of quotients).

The definitions, the axiomatic formulations and the properties of this quotient ringUcan be found in [1].

In any case, whenRis a prime ring, all that we need aboutUis that 1) RU;

2) U is a prime ring;

3) The center of U, denoted by C, is a field which is called the extended centroid ofR.

We make also a frequent use of the theory of generalized polynomial identities and differential identities (see [1], [12], [16], [19]). In particular we need to recall that whenRis prime andIa non-zero right ideal ofR, thenI;IRandIUsatisfy the same generalized polynomial identities [4].

In [13] T.K. Lee extended the definition of a generalized derivation as follows; by a generalized derivation we mean an additive mappingg:I!U such thatg(xy)ˆg(x)y‡xd(y) for allx;y2I, whereIis a dense right ideal of Randd is a derivation fromIintoU. Moreover Lee also proved that every generalized derivation can be uniquely extended to a generalized derivation ofUand thus all generalized derivations ofRwill be implicitly assumed to be defined on the wholeUand obtained the following results:

THEOREM. (Theorem 4, [13]). Every generalized derivation g on a dense right idealof a semiprime ring R can be uniquely extended to U and assumes the form g(x)ˆax‡d(x);for some a2U and a derivation d on U.

More detail about generalized derivations can be found in [13], [10], [17].

REMARK3. It is well known that every derivation ofRcan be uniquely extended to a derivation ofU[16]. Moreover sinceRis prime ring , we may assumeKCand so for anya2K one hasd(a1)2C.

We will use the following notation:

f(x1;. . .;xn)ˆax1x2. . .xn‡X

s6ˆ1

asxs(1)xs(2). . .xs(n)

for somea;as2Cand moreover we denote byf^d(x1;. . .;xn) the polynomial obtained fromf(x1;. . .;xn) by replacing each coefficientaswithd(as). Thus we writed(f(r1;. . .;rn))ˆf^d(r1;. . .;rn)‡P

i f(r1;. . .;d(r_i);. . .;rn), for all r₁;r₂;. . .;r_n2R:

REMARK4. We will also write a multilinear polynomialf(x1;. . .;xn) as follows:

f(x1;. . .;xn)ˆX

i

t_i(x1;. . .;x_{i 1};x_i‡1;. . .;xn)x_i

wheret_i are multilinear polynomials inn 1 variables, andx_i never ap- pears in any monomials oft_i.

Before the beginning of our proofs, we would like to recall Wang's re- sults, more precisely we refer to Theorem 2 and Theorem 4 in [21]. All that we need here is to remind the conclusions contained in [21] in the casegis an inner generalized derivation ofR. We summarize these reduced results in the following Facts 1,2 and 3:

FACT1. Let a;b2R such that(af(x₁;. . .;x_n)‡f(x₁;. . .;x_n)b)^mˆ0, for all x₁;. . .;x_n 2R. Then either aˆ b2Z(R)or f(x₁;. . .;x_n)is central valued on R.

FACT 2. As a specialsimple case of Fact 1 we have that if

f(x1;. . .;xn) is a non-central multilinear polynomial over K such

that (af(x1;. . .;xn))^mˆ0, for al l x1;. . .;xn2R, then aˆ0.

FACT3. Let R be a prime ring and a;b2R, and m1be a fixed in- teger. Suppose that f(x1;. . .;xn)is a multilinear polynomial over K which is not vanishing on R, such that(af(x₁;. . .;xn)‡f(x₁;. . .;xn)b)^m2Z(R) for all x₁;. . .;x_n 2R. Then one of the following holds:

(i) f(x₁;. . .;x_n)is central-valued on R;

(ii) aˆ b2Z(R);

(ii) R satisfies s4;

(iv) a;b2Z(R)and f(x1;. . .;xn)is power centralvalued on R.

3. The nilpotent case.

We begin with the following:

LEMMA 1. Let R be a prime K-algebra, I a non-zero right ideal of R, a2R, m1, a fixed integer, f(x1;. . .;xn)a multilinear polynomial over K such that (af(r₁;. . .;r_n))^mˆ0, for all r₁;. . .;r_n2I. Then one of the following holds:

(i) [f(x1;. . .;xn);xn‡1]xn‡2 is an identity for I;

(ii) aIˆ0.

PROOF. Suppose that conclusions (i) and (ii) do not occur. We prove that, in this case, we get a contradiction. Letb₁;. . .;b_n‡2;w2Isuch that

[f(b₁;. . .;b_n);b_n‡1]b_n‡26ˆ0 and aw6ˆ0 (1)

SinceaI6ˆ0, there exists a non-central elementuofIsuch thatau6ˆ0.

Hence (af(ux₁;. . .;uxn))^mis a non-trivial generalized polynomial identity forR. ThusRis a GPI-ring, so thatRChas a non-zero socleHwith non- zero right ideal JˆIH. Moreover His a simple ring with minimal right ideals (see [19]). Note thatJˆJHandJsatisfies the same basic conditions asI. ReplacingRbyH;IbyJwe have thatRis a simple ring with minimal right ideals, moreoverRis equals to its own socle andIRˆI. Recall that, by Remark 2, we have

f(x₁;. . .;x_n)ˆX

i

t_i(x₁;. . .;x_{i 1};x_i‡1;. . .;x_n)x_i:

Since R is a simple ring with minimal right ideals, it is a right-semi- simple ring. Of course it is also a left-semisimple ring, so that R is a semisimple noetherian ring. Therefore any right (left) ideal ofRis finitely generated. Moreover, sinceRis regular there existse²ˆe2IRsuch that

eRˆwR‡^n‡2P

iˆ1b_iR such that, ewˆw;eb_iˆb_i;iˆ1;. . .;n‡2. In parti- cular for allr₁; ::;r_n2Rand for alliˆ1; ::;nwe have

(af(er₁;. . .;er_i(1 e);. . .;er_n))^mˆ(at_i(er₁;. . .;er_n)er_i(1 e))^mˆ0:

Hence for alliˆ1; ::;n

((1 e)at_i(er1;. . .;ern)er_i)^m‡1ˆ0

that is (1 e)at_i(er₁;. . .;er_n)eR is a nil right ideal of bounded in- dex. Thus, by a well known result which is attribuited to Levitzki (for a proof see [6] and pp. 1-2 in [9]), (1 e)aeti(er1;. . .;ern)eriˆ0 for any r1;. . .;rn2R. By the primeness of R it follows that (1 e)aet_i(er1;. . .;ern)eˆ0 for all iˆ1; ::;n. Thus the left ideal Re satisfies the generalized identity (1 e)aet_i(x₁;. . .;x_n). By main theorem in [5] we have that (1 e)ae(Re)t_i(Re;. . .;Re)ˆ(0).

Therefore, again by the primeness of R, we get that either

(1 e)aeˆ0 or ti(x1;. . .;xn)xi is an identity for eR for all

iˆ1; ::;n.

Firstly we assume that (1 e)aeˆ0. Then (eaef(ex₁e;. . .;ex_ne)^m is an identity for R. In view of Fact 2, we have that either eaeˆ0 or

f(x₁;. . .;x_n) is central for eRe. Second possibility gives that

[f(er1;. . .;ern);ern‡1]ern‡2ˆ0. In particular

0ˆ[f(eb1;. . .;ebn);ebn‡1]ebn‡2ˆ[f(b1;. . .;bn);bn‡1]bn‡26ˆ0;

a contradiction. If eaeˆ0, then 0ˆaeˆaewˆaw, a contradiction again.

Finally we assume thatti(er1;. . .;ern)eriˆ0 for anyr1;. . .;rn2Rand for alliˆ1; ::;n. Then, in particular, we get the contradiction

06ˆ[f(eb₁;. . .;eb_n);eb_n‡1]eb_n‡2ˆh X

i

t_i(eb₁;. . .;eb_n)eb_i;b_n‡1i

b_n‡2ˆ0:

p LEMMA 2. Let R be a prime ring, a;b2R, m1 a fixed integer, I a non-zero right idealof R, f(x1;. . .;xn)a multilinear polynomial over K such that(af(r1;. . .;rn)‡f(r1;. . .;rn)b)^mˆ0, for all r1;. . .;rn 2I.

Then either[f(x₁;. . .;x_n);x_n‡1]x_n‡2is an identity for I, or(a‡b)Iˆ0 and[b;I]Iˆ0.

PROOF. If b2C then the proof is finished by Lemma 1. So we may assume thatb2=C. First aim is to show thatRis a GPI-ring. Letu2Isuch

thatfau;ugare linearly C-independent. Then

(af(ux1;. . .;uxn)‡f(ux1;. . .;uxn)b)^m (2)

is a nontrivial generalized polynomial identity forR, as (af(ux1;. . .;uxn))^m occurs nontrivially in (2). So Ris a GPI-ring. Let nowauˆau for some a2C. ThenRsatisfies

(af(ux₁;. . .;ux_n)‡f(ux₁;. . .;ux_n)b)^m (3)

Iffbu;ugare C-independent, then (f(ux1;. . .;uxn)b)^moccurs nontrivially in (3). ThenRis a GPI-ring. If buˆbu, for some b2C, then the coeffi- cients which occur in (3) are only fu;bg. If uˆmb for somem2C, then, sinceb2=C,Ris GPI-ring. On the other hand, if fu;bgare linearly C-in- dependent, then (3) is again a non-trivial generalized polynomial identity forR. Therefore, again by [19],RChas a non-zero socleHwith non-zero right idealJˆIH. As aboveHis a simple ring with minimal right ideals, moreover we have thatJˆJHandJsatisfies the same basic conditions as Iin view of [16]. ReplacingRby H;IbyJwe have thatRis simple and equals to its own socle andIRˆI. Notice that if (a‡b)Iˆ0 then, from the assumption (af(r1;. . .;rn)‡f(r1;. . .;rn)b)^mˆ0 for allr1;. . .;rn 2I, it follows that [b;f(r1;. . .;rn)]^m ˆ0 for all r1;. . .;rn2Iand by [2] we have that either [b;I]Iˆ0 or there exists an idempotent elemente2Hsuch thatICˆeRC andf(x₁; ::;x_n) is central valued oneRCe. In this last case we have [f(er₁e;. . .;er_ne);er_n‡1e]eˆ0, for all r₁; ::;r_n‡12R, that is

[f(er1;. . .;ern);ern‡1]eˆ0. In other words I satisfies the polynomial

identity [f(x1;. . .;xn);xn‡1]xn‡2and we are done.

Now we assume that the conclusion of the lemma doesn't hold. Then there existw;b₁;. . .b_n‡22Isuch that

(a‡b)w6ˆ0 and [f(b₁;. . .;b_n);b_n‡1]b_n‡26ˆ0:

As in Lemma 1, since R is a simple regular ring, we write P

n‡2

iˆ1b_iR‡wRˆeRfor some idempotent elemente2R. Moreovere2IR

and eb_iˆb_i foriˆ1;. . .;n‡2 and ewˆw. By the hypothesis we have

that, for alliˆ1; ::;n,Rsatisfies

(af(ex1;. . .;exi(1 e);. . .exn)‡f(ex1;. . .;exi(1 e);. . .exn)b)^m and by calculations it follows thatRsatisfies

(at_i(ex1;. . .;exn)ex_i(1 e)‡t_i(ex1;. . .;exn)ex_i(1 e)b)^m (4)

for alliˆ1; ::;n. Left multiplying (4) by (1 e), we get

0ˆ(1 e)(at_i(er₁;. . .;er_n)er_i(1 e))^m 8r₁; ::;r_n2R:

It means that, for allr1; ::;rn2Rand for alliˆ1; ::;n, ((1 e)(at_i(er1;. . .;ern)eR)^m‡1ˆ(0):

By the above cited result of Levitzki (see again [6] and pp.1-2 in [9]) we have thatRsatisfies (1 e)aet_i(ex₁;. . .;ex_n)ex_ifor alliˆ1; ::;n.

Here we repeat the same argument in Lemma 1: by the primeness ofR it follows thatRsatisfies (1 e)aet_i(ex1;. . .;exn)efor alliˆ1; ::;n. Thus the left idealResatisfies the generalized identity (1 e)aet_i(x1;. . .;xn). By main theorem in [5] we have that (1 e)ae(Re)t_i(Re;. . .;Re)ˆ(0). It fol- lows that either (1 e)aeˆ0 ort_i(x₁;. . .;x_n)x_iis an identity foreRfor all iˆ1; ::;n.

If second case holds, then in particular we get the contradiction 06ˆ[f(eb1;. . .;ebn);ebn‡1]ebn‡2ˆ

h X

i

t_i(eb₁;. . .;eb_{i 1};eb_i‡1;. . .;eb_n)eb_i;b_n‡1i

b_n‡2ˆ0:

So we haveaeˆeae. Now right multiplying (4) byewe get (t_i(er₁;. . .;er_n)er_i(1 e)b)^meˆ0

for allr1; ::;rn2Randiˆ1; ::;n, that is

((1 e)bet_i(er1;. . .;ern)eR)^m‡1ˆ(0):

Once again by Levitzki's theorem we have thatRsatisfies the generalized identity (1 e)bet_i(ex₁;. . .;ex_n)ex_ifor alliˆ1; ::;n.

Thus the left ideal Re satisfies the generalized identity (1 e)bet_i(x1;. . .;xn). By main theorem in [5] we have that (1 e)be(Re)t_i(Re;. . .;Re)ˆ(0). It follows that either (1 e)beˆ0 or t_i(x₁;. . .;x_n)x_i is an identity foreRfor alliˆ1; ::;n.

As above, in this last case we get the contradiction 06ˆ[f(eb1;. . .;ebn);ebn‡1]ebn‡2ˆ

ˆh X

i

t_i(eb1;. . .;eb_{i 1};eb_i‡1;. . .;ebn)eb_i;bn‡1

ibn‡2ˆ0:

Hence it follows that (1 e)beˆ0. Therefore the finite dimensional simple central algebraeRCesatisfies (af(x₁;. . .;x_n)‡f(x₁;. . .;x_n)b)^m. Then by Fact 1, eithereaeˆ ebe2Corf(ex₁e;. . .;ex_ne) is central. In the last case we have [f(er1;. . .;ern);ern‡1]ern‡2ˆ[f(r1;. . .;rn);rn‡1]rn‡26ˆ0, a contradiction. Finally if aeˆeaeˆ ebeˆ be, then 0ˆ(a‡b)eˆ

ˆ(a‡b)ewˆ(a‡b)w6ˆ0, which is again a contradiction. p

THEOREM1. Let K be a commutative ring with unity, R a prime K- algebra with extended centroid C and g a non-zero generalized derivation of R. Suppose that f(x₁;. . .;xn)is a multilinear polynomial over K and I is a non-zero right idealof R and m1, a fixed integer.

If(g(f(r₁;. . .;r_n)))^mˆ0, for all r₁;. . .;r_n2I, then one of the following holds:

(i) [f(x1;. . .;xn);xn‡1]xn‡2 is an identity for I;

(ii) g(a)ˆax for all x2R, where a2U such that aIˆ0;

(iii) g(x)ˆax [q;x]for all x2R, where a;q2U such that aIˆ0 and[q;I]Iˆ0.

PROOF. As remarked above, every generalized derivationgon a dense right ideal of R can be uniquely extended to U and assumes the form g(x)ˆax‡d(x), for some a2U and derivation d onU. Therefore, for u2I;Usatisfies the following differential identity

(af(ux₁;. . .;ux_n)‡d(f(ux₁;. . .;ux_n)))^mˆ0 Suppose that [f(x₁;. . .;x_n);x_n‡1]x_n‡2is not identity forI.

Ifdˆ0, then (af(x1;. . .;xn))^mˆ0 for allx1;. . .;xn2I:Hence by the initial hypothesis and Lemma 1 we getg(x)ˆaxfor allx2RandaIˆ0.

So the conclusion (ii) holds.

We may assume thatd6ˆ0. Thus we have thatIsatisfies af(x₁;. . .;x_n)‡f^d(x₁;. . .;x_n)‡X

i

f(x₁;. . .;d(x_i);. . .;x_n)_m :

In light of Kharchenko's theory [12], we divide the proof into two case:

CASE1. Ifdis inner derivation induced by the elementq2U, that is d(x)ˆqx xqfor allx2U. Thusg(x)ˆax‡d(x)ˆ(a‡q)x xqandU satisfies

(a‡q)f(ux1;. . .;uxn) f(ux1;. . .;uxn)q_m :

Since [f(x1;. . .;xn);xn‡1]xn‡2 is not identity forI, then by Lemma 2 we have that 0ˆ(a‡q q)IˆaIand [q;I]Iˆ0.

CASE2. Let nowdan outer derivation ofU. SinceIandIUsatisfies the same differential identities,

af(x₁;. . .;x_n)‡d(f(x₁;. . .;x_n))_m is an identity forIU, that is, for anyu2I,

af(ux1;. . .;uxn)‡d(f(ux1;. . .;uxn))_m

is an identity forU. ThusUsatisfies the following af(ux1;. . .;uxn)‡f^d(ux1;. . .;uxn)‡X

i

f(ux1;. . .;d(u)xi‡ud(xi);. . .;uxn)_m : Sincedis an outer derivation, by Kharchenko's results in [12],Usatisfies the identity

af(ux1;. . .;uxn)‡f^d(ux1;. . .;uxn)‡X

i

f(ux1;. . .;d(u)xi‡uyi;. . .;uxn)_m : In particular taking xiˆ0 it is easy to see that U satisfies the blended componentf(ux1;. . .uyi;. . .;uxn)^m, that isI satisfies f(x1;. . .;yi;. . .;xn)^m. SinceIsatisfies a non-trivial polynomial identity, then there exists an idem- potente2socle(RC) such thatIˆeRC (see Proposition in [15]). HenceR satisfiesf(ex1;. . .;exn)^mthat isRsatisfies the identityef(ex1;. . .;exn)^m.

Then by main theorem in [5] we have that R satisfies ef(ex₁;. . .;ex_n)ex_n‡1 that is f(x₁;. . .;y_i;. . .;x_n)x_n‡1 is satisfied by

IˆeRC, a contradiction. p

4. The power central case.

At first we need to fix the following:

LEMMA3. Let R be a prime ring, I a non-zero right idealof R and m1 a fixed integer. If f(x1;. . .;xn) is a multilinear polynomial over K such that f(r₁;. . .;r_n)^m2Z(R) for all r₁;. . .;r_n2I, then either

f(x₁;. . .;x_n)x_n‡1 is an identity for I or f(x₁;. . .;x_n) is power central

valued on R.

PROOF. Suppose thatf(r1;. . .;rn)^m2Z(R) for anyr1;. . .;rn2I. Since Isatisfies a non-trivial polynomial identity, then there exists an idempotent e2socle(RC) such that IˆeRC (see Proposition in [15]). Thus, since [f(r₁;. . .;r_n)^m;r]ˆ0 for allr₁;. . .;r_n2Iandr2R, we have that for any r1;. . .;rn‡12R

0ˆ[f(er1;. . .;ern)^m;rn‡1(1 e)]ˆf(er1;. . .;ern)^mrn‡1(1 e):

SinceRis prime, either (1 e)ˆ0 orf(er1;. . .;ern)^mˆ0. In the first caseeˆ1 and IˆRC. Thusf(x₁;. . .;x_n) is power central valued onRC and so also on R. On the other hand, if f(er₁;. . .;er_n)^mˆ0, then ef(er1;. . .;ern)^mˆ0 and by main theorem in [5]ef(er1;. . .;ern)eRˆ0, for

allr1;. . .;rn 2R. Hencef(x1;. . .;xn)xn‡1is an identity forIˆeRCand we

are done. p

THEOREM2. Let K be a commutative ring with unity, R a prime K- algebra, with extended centroid C, g a non-zero generalized derivation of R.

Suppose that f(x1;. . .;xn)is a multilinear polynomial over K, I is a non- zero right idealof R and m1, a fixed integer such that g(f(r1;. . .;rn))^m2Z(R) for all r1;. . .;rn2I. If there exist a1;. . .;an2I such that g(f(a₁;. . .;a_n))^m6ˆ0, then one of the following holds:

(i) f(x1;. . .;xn)xn‡1 is an identity for I;

(ii) f(x₁;. . .;xn)is centralvalued on R;

(iii) g(x)ˆax for a2C and f(x₁;. . .;x_n)is power centralvalued on R;

(iv) R satisfies s₄.

PROOF. Write again g(x)ˆax‡d(x), for suitable a2U andda de- rivation ofR. Since (af(x1;. . .;xn)‡d(f(x1;. . .;xn)))^m is a central differ- ential identity forI, by Theorem 1 in [3]Ris a PI-ring and soRCis a finite dimensional central simple C-algebra. By Wedderburn-Artin theorem RCM_k(D) for somek1 andDa finite-dimensional central division C- algebra. By Theorem 2 in [16] (af(x1;. . .;xn)‡d(f(x1;. . .;xn)))^m2Cfor

allx1;. . .;xn2IC. Without loss of generality we may replaceRwithRC

and assume that RˆM_k(D). Let s1;. . .;sn be arbitrary elements of I.

ThenRsatisfies the following central differential identity in the variables XandY:

"

af(s₁X;s₂;. . .;s_n)‡f^d(s₁X;s₂;. . .;s_n)‡f(d(s₁)X‡s₁d(X);s₂;. . .;s_n)‡

‡X

i2

f(s1X;. . .;d(si);. . .;sn)_m

;Y

# : If d is an outer derivation, fix d(X)ˆy, Xˆ0 and obtain f(s₁r;s₂;. . .;s_n)^m2C, for alls₁;. . .;s_n 2Iandr2R. By Lemma 3 either f(x₁;. . .;x_n)x_n‡1is an identity forIorf(x₁;. . .;x_n) is power central valued onR. In the first case we are done.

Hence we assume thatf(x1;. . .;xn)^m is central valued on R. In parti- cular, iff(r1;. . .;rn)^mˆ0 for allr1;. . .;rn 2Ithen we obtain as a reduc- tion of Lemma 3 that f(x₁;. . .;x_n)x_n‡1 is an identity for I. In the either case there exists b₁;. . .;b_n2Isuch that 06ˆf(b₁;. . .;b_n)^m2C, that isI contains an invertible element ofR, and soIˆR. Hence we conclude by Wang's theorem in [21].

Suppose now thatdis an inner derivation, sayd(x)ˆ[q;x]ˆqx xq.

Let F be a maximal subfield of D, so that Mk(D)_CFMt(F) where tˆk[F:C]. Hence the derivationdcan be extended toM_k(D)_CFand (af(x₁;. . .;x_n)‡d(f(x₁;. . .;x_n)))^m2Z(M_t(F)), for any x₁;. . .;x_n2IF (Lemma 2 in [14] and Proposition in [18]). Therefore we may assume that RMt(F). Notice that if tˆ1;2 then we have respectively that either RForRM2(F). In any caseRsatisfies the standard identitys4and we are done.

Here we assume thatt3. Denotee_ijthe usual matrix unit with 1 in the (i;j)-entry and zero elsewhere. Since there exists a set of matrix units that contains the idempotent generator of a given minimal right ideal, we ob- serve that any minimal right ideal is part of a direct sum of minimal right ideals adding toR. In light of this and applying Proposition 5 on page 52 in [11], we may assume that any minimal right ideal ofRis a direct sum of minimal right ideals, each of the forme_iiR.

Moreover we know that the right ideal I has a number of uniquely determinated simple components: they are minimal right ideals ofRandI is their direct sum. In light of this argument, we may writeIˆeRfor some eˆP^l

iˆ1e_ii and l2 f1;2; :::;tg, that is IˆeRˆ(e₁₁R‡. . .‡e_llR), where t3 andlt.

DenoteqˆP

r;sq_rse_rs,aˆP

r;sa_rse_rs, forq_rs;a_rs2F.

Iff(x1;. . .;xn) is not an identity forI, then by Lemma 3 in [2], for any

il,j>l;the elemente_ijfalls in the additive subgroup ofRCgenerated by all valuations off(x₁;. . .;x_n) inI. Since the matrix (ae_ij‡qe_ij e_ijq)^m has rank2, then it is not central. Therefore (ae_ij‡qe_ij e_ijq)^mˆ0 and right multiplying by eii we have ( eijq)^meiiˆ0. This means that qjiˆ0 for any il and for any j>l, that is qII. Moreover from e_jj(ae_ij‡qe_ij e_ijq)^mˆ0 we also get e_jj((a‡q)e_ij)^mˆ0, that is a_jiˆ0, and aII. All this implies that g(I)I. Since 06ˆg(f(a₁;. . .;a_n))^m2 2I\F, it is invertible andIˆR. Also in this case we conclude thanks to Theorem 4 in [21].

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Manoscritto pervenuto in redazione il 19 marzo 2007.