Generalized Derivations on (Semi-)Prime Rings and Noncommutative Banach Algebras

Generalized Derivations on (Semi-)Prime Rings and Noncommutative Banach Algebras

FENGWEI(*) - ZHANKUIXIAO(*)

ABSTRACT- We first give several polynomial identities of semiprime rings which make the additive mappings appearing in the identities to be generalized de- rivations. Then we study some pairs of generalized Jordan derivations with power values on prime rings. Letm;nbe fixed positive integers,Rbe a non- commutative2(m‡n)!-torsion free prime ring with the centerZandm;nbe a pair of generalized Jordan derivations onR. If m(x^m)xⁿ‡xⁿn(x^m)2 Zfor all x2 R, thenmandnare left (or right) multipliers. In particular, ifm;nare a pair of derivations onRsatisfying the same assumption, thenmˆnˆ0. Then ap- plying these purely algebraic result we obtain several range inclusion results of pair of derivations on Banach algebras.

0. Introduction.

LetRbe an associative ring. An additive mappingm:R ! Ris called a generalized derivationof Rif there exists a derivationd ofRsuch that m(xy)ˆm(x)y‡xd(y) for all x;y2 R, where d is called the associated derivation ofm. Ifm is a generalized derivation ofRwith the associated derivationd, thenmsatisfies the following relations

m(xⁿ)ˆm(x)x^{n 1}‡X^{n 1}

jˆ1

x^{n j}d(x)x^{j 1} (0:1)

and

m(xⁿ)ˆm(x^{n m})x^m‡x^{n m}d(x^m) (0:2)

(*) Indirizzo degli A.: Department of Applied Mathematics, Beijing Institute of Technology, Beijing, 100081, P. R. China.

E-mail: daoshuo@bit.edu.cn zhkxiao@bit.edu.cn 2000Mathematics Subject Classification. 16W25, 16N60.

for all x2 R, where m;n are positive integers with 1mn 1. It seems natural to ask under what additional assumptions the converse is true. More precisely, under what additional assumptions an additive mappingm, which maps a ringRinto itself, satisfying one of the relations (0:1) and (0:2) is a generalized derivation. Bridges and Bergen [4], Vuk- mann and Kosi-ulbl [25] have considered the similar problems for the case of derivations on rings. In this paper we will give affirmative an- swers to just mentioned questions under mild conditions.

Inspired by the relations (0:1) and (0:2) we consider much more common generalized forms of skew centralizing mappings involved pair of (general- ized-)Jordan derivations with power values on a prime ring R. Precisely speaking, the generalized forms are

(a) m(x^m)xⁿ‡xⁿn(x^m)2 Z;

(b) d(x^m)xⁿ‡xⁿg(x^m)2 Z;

(c) x^md(x)xⁿ‡x^pg(x)x^q 2 Z,

wherem;n;p;qare positive integers,m;nare a pair of generalized Jordan derivations onR,d;gare a pair of Jordan derivations onRandZis the center of R. In fact, some similar identities with (generalized-)deriva- tions have been studied by many people in various way, see [1], [9], [11], [14], [24], [27] and [28]. In the present paper we prove the following re- sults. LetRbe a noncommutative 2(m‡n)!-torsion free prime ring . If (a) holds for all x2 R, then m and n are left (or right) multipliers. In particular, if (b) holds for all x2 R, then dˆgˆ0. If (c) holds for all x2 R, thendˆgˆ0. All these results generalize many existed conclu- sions in this field.

On the other hand, many authors use pure algebraic techniques (especially, the techniques concerning (semi-)prime rings) to study the range inclusion problem of derivations on Banach algebras, which are related to the well known noncommutative Singer-Wermer conjecture.

Various partial answers to this conjecture are obtained. Most results in the field of range inclusion deal with one derivation, while pair of deri- vations on Banach algebras was more less considered. In this paper we apply the algebraic results in the previous paragraph to discuss pairs of derivations on Banach algebras. LetAbe a noncommutative Banach al- gebra with the Jacobson radicalrad(A) andd;gbe a pair of derivations on A. Ifd(x^m)xⁿ‡xⁿg(x^m)2 Z_Afor allx2 Aand fixed positive integersm;

n, then d(A)rad(A) andg(A)rad(A). The case of a pair of Jordan derivations on Banach algebras is our next goal. Let A be a non- commutative Banach algebra with the Jacobson radicalrad(A) andd;gbe

a pair of continuous Jordan derivations on A. If d(x^m)xⁿ‡xⁿg(x^m)2 2rad(A) for all x2 A and fixed positive integers m;n, then d(A)rad(A) andg(A)rad(A).

1. Some Additive Mappings are Generalized Derivations.

Throughout this sectionRalways denotes an associative ring. IfRhas the identity elemente, wecan adopt the convention thatx⁰ˆefor allx2 R n f0g.

Recall that a ringRisprimeif for alla;b2 R,aRbˆ0 implies that either aˆ0 orbˆ0, and issemiprimein caseaRaˆ0 implies thataˆ0. An ad- ditive mappingd:R ! Ris said to be aderivationifd(xy)ˆd(x)y‡xd(y) for all forx;y2 Rand is called aJordan derivationifd(x²)ˆd(x)x‡xd(x) for allx2 R. Every derivation is a Jordan derivation, but the converse is in general not true. A classical result of Herstein [12] asserts that any Jordan derivation on a 2-torsion free prime ring is a derivation. Cusack [6] extended Herstein's result to the case of semiprime rings and proved that every Jordan derivation on a 2-torsion free semiprime ring is a derivation. Letmbe a gen- eralized derivation ofRwith the associated derivationd. Thenmsatisfies the relation (0:1). We find that under additional assumptions, the converse is also true for a class of additive mappings on semiprime rings.

THEOREM 1.1. Let n>1 be a fixed positive integer and Rbe a n!- torsion free semiprime ring with the identity element e. Suppose that there exist additive mappingsm:R ! R, d:R ! Rsuch that

m(xⁿ)ˆm(x)x^{n 1}‡X^{n 1}

jˆ1

x^{n j}d(x)x^{j 1}

for all x2 R. Thenmis a generalized derivation onRwith the associated derivation d.

PROOF. The assumption implies that m(xⁿ) m(x)x^{n 1} X^{n 1}

jˆ1

x^{n j}d(x)x^{j 1}ˆ0 (1:1)

for allx2 R. By (1:1) we immediately getd(e)ˆ0. Substitutingx‡lyforx in (1:1) we obtain

lP₁(x;y)‡l²P₂(x;y)‡ ‡lⁿP_n(x;y)ˆ0;

wherel2Z,x;y2 R,Pi(x;y) denotes the sum of terms involvingifactors ofyin the expansion of

m((x‡ly)ⁿ) m(x‡ly)(x‡ly)^{n 1} X^{n 1}

jˆ1

(x‡ly)^{n j}d(x‡ly)(x‡ly)^{j 1}ˆ0:

By [5, Lemma 1] it follows that

(1:2) P₁(x;y)ˆm(x^{n 1}y‡x^{n 2}yx‡ ‡yx^{n 1}) m(y)x^{n 1} m(x)(x^{n 2}y‡x^{n 3}yx‡ ‡yx^{n 2})

X^{n 1}

jˆ1

(x^{n j 1}y‡ ‡yx^{n j 1})d(x)x^{j 1} X^{n 1}

jˆ1

x^{n j}d(y)x^{j 1} X^{n 1}

jˆ1

x^{n j}d(x)(x^{j 2}y‡x^{j 3}yx‡ ‡yx^{j 2})ˆ0 for allx;y2 R. Takingxˆeinto (1:2) leads to

nm(y) m(y) (n 1)m(e)y (n 1)d(y)ˆ0 for ally2 R. Note thatRisn!-torsion free. So we have

m(y)ˆm(e)y‡d(y) (1:3)

for ally2 R. Replacingybyxⁿin (1:3) yields m(xⁿ)ˆm(e)xⁿ‡d(xⁿ) for allx2 R. This implies that

m(e)xⁿ‡d(xⁿ)ˆm(x)x^{n 1}‡X^{n 1}

jˆ1

x^{n j}d(x)x^{j 1} (1:4)

for allx2 R. Combining (1:4) with (1:3) we get d(xⁿ)ˆXⁿ

jˆ1

x^{n j}d(x)x^{j 1} (1:5)

for allx2 R. Substitutingx‡lyforxin (1:5) leads to lP₁(x;y)‡l²P₂(x;y)‡ ‡lⁿP_n(x;y)ˆ0;

wherel2Z,x;y2 R,Pi(x;y) denotes the sum of terms involvingifactors ofyin the expansion of

d((x‡ly)ⁿ) Xⁿ

jˆ1

(x‡ly)^{n j}d(x‡ly)(x‡ly)^{j 1}ˆ0:

By [5, Lemma 1] again we obtain P₂(e;y)ˆn(n 1)

2 d(y²) Xⁿ

jˆ1

(n j)yd(y) Xⁿ

jˆ1

(j 1)d(y)yˆ0 (1:6)

for ally2 R. Note thatRisn!-torsion free. So d(y²)ˆd(y)y‡yd(y)

for ally2 R. This shows thatdis a Jordan derivation onRand hence is a derivation onR. Thus the identity (1:3) shows thatmis a generalized de- rivation onRwith the associated derivationd. p

As a direct corollary of Theorem 1.1 we have

COROLLARY1.2 [25, Theorem 1]. Letn >1be a fixed positive integer

and Rbe a n!-torsion free semiprime ring with the identity elemente.

Suppose that there exists an additive mappingd:R ! Rsuch that d(xⁿ)ˆXⁿ

jˆ1

x^{n j}d(x)x^{j 1} for all x2 R. Then d is a derivation onR.

According to the definition of generalized derivation, ifmis a general- ized derivation ofRwith the associated derivationd, thenmalso satisfies another relation (0:2). Analogously, under mild conditions, the converse also holds for a class of additive mappings on semiprime rings.

THEOREM 1.3. Let n>1 be a fixed positive integer and Rbe a n!- torsion free semiprime ring with the identity element e. Suppose that there exist additive mappingsm:R ! Rand d:R ! Rsuch that

m(xⁿ)ˆm(x^{n m})x^m‡x^{n m}d(x^m)

for all x2 R, where m is a positive integer with1mn 1. Thenmis a generalized derivation onRwith the associated derivation d.

PROOF. The assumption implies that

m(xⁿ) m(x^{n m})x^m x^{n m}d(x^m)ˆ0 (1:7)

for allx2 R. Then it is easy to see thatd(e)ˆ0. Substitutingx‡lyforx in (1:7) we obtain

lP1(x;y)‡l²P2(x;y)‡ ‡lⁿPn(x;y)ˆ0;

wherel2Z,x;y2 R,Pi(x;y) denotes the sum of terms involvingifactors of y in the expansion of m((x‡ly)ⁿ) m((x‡ly)^{n m})(x‡ly)^m

(x‡ly)^{n m}d((x‡ly)^m)ˆ0. It follows from [5, Lemma 1] that (1:8) P₁(x;y)ˆm(x^{n 1}y‡x^{n 2}yx‡ ‡yx^{n 1})

m(x^{n m 1}y‡ ‡yx^{n m 1})x^m m(x^{n m})(x^{m 1}y‡x^{m 2}yx‡ ‡yx^{m 1}) (x^{n m 1}y‡ ‡yx^{n m 1})d(x^m)

x^{n m}d(x^{m 1}y‡x^{m 2}yx‡ ‡xyx^{m 2}‡yx^{m 1})ˆ0 for allx;y2 R. Takingxˆeinto (1:8) yields

nm(y) (n m)m(y) mm(e)y md(y)ˆ0 for ally2 R. Note thatRisn!-torsion free. So

m(y)ˆm(e)y‡d(y) (1:9)

for ally2 R. Combining (1:9) with (1:7) we get d(xⁿ)ˆd(x^{n m})x^m‡x^{n m}d(x^m) (1:10)

for allx2 R. Substitutingx‡lyforxin (1:10) leads to lP1(x;y)‡l²P2(x;y)‡ ‡lⁿPn(x;y)ˆ0;

where l2Z, x;y2 R, P_i(x;y) denotes the sum of terms involving i factors of y in the expansion ofd((x‡ly)ⁿ) d((x‡ly)^{n m})(x‡ly)^m

(x‡ly)^{n m}d((x‡ly)^m)ˆ0. By [5, Lemma 1] again we have P2(e;y)ˆn(n 1)

2 d(y²) (n m)(n m 1)

2 d(y²)

(1:11)

(n m)md(y)y (n m)myd(y) m(m 1)

2 d(y²)ˆ0 for ally2 R. In view of the torsion free fact ofR, we see that

d(y²)ˆd(y)y‡yd(y)

for ally2 R. This shows thatdis a Jordan derivation onRand hence is a derivation. Thus the identity (1:9) proves thatmis a generalized derivation

onRwith the associated derivationd. p

As an immediate consequence of Theorem 1.3 we have

COROLLARY1.4. Let n>1 be a fixed positive integer andRbe a n!- torsion free semiprime ring with the identity element e. Suppose that there

exists an additive mapping d:R ! Rsuch that d(xⁿ)ˆd(x^{n m})x^m‡x^{n m}d(x^m)

for all x2 R, where m is a positive integer with1mn 1. Then d is a derivation onR.

An additive mapping m:R ! Ris called a generalized Jordan deri- vation of R if there exists a Jordan derivation d of R such that m(x²)ˆm(x)x‡xd(x) for allx2 R, wheredis called theassociated Jordan derivation of m. Obviously, any generalized derivation is a generalized Jordan derivation. But, the converse statement is not necessarily true. It has been proved that any generalized Jordan derivation on a 2-torsion free semiprime ring is a generalized derivation [27]. Thus, ifmis a generalized Jordan derivation on a 2-torsion free semiprime ring R, then m(xyx)ˆ

ˆm(x)yx‡xd(yx) for allx;y2 R. Under mild assumptions, the converse also holds for a class of additive mappings on a 2-torsion free semiprime ring. But, the condition thatRcontains the identity elementeis not ne- cessary here, which is different form the additional assumption of Theorem 1.1 and that of Theorem 1.3.

THEOREM1.5. LetRbe a2-torsion free semiprime ring,m:R ! Rbe an additive mapping and d be a Jordan derivation of R. If m(xyx)ˆ

ˆm(x)yx‡xd(yx)for all x;y2 R, thenmis a generalized derivation with the associated derivation d.

PROOF. The linearization of the assumption m(xyx)ˆm(x)yx‡xd(yx) (1:12)

gives

m(xyz‡zyx)ˆm(x)yz‡xd(yz)‡m(z)yx‡zd(yx) (1:13)

for allx;y;z2 R. Takingzˆx²into (1:13) we obtain

m(xyx²‡x²yx)ˆm(x)yx²‡xd(yx²)‡m(x²)yx‡x²d(yx) (1:14)

for allx;y2 R. Substitutingxy‡yxforyin (1:12) yields

m(x²yx‡xyx²)ˆm(x)xyx‡m(x)yx²‡xd(xyx)‡xd(yx²) (1:15)

for allx;y2 R. From (1:14) and (1:15) we have A(x)yxˆ0 (1:16)

for allx;y2 R, whereA(x) denotesm(x²) m(x)x xd(x). It is sufficient to prove thatA(x)ˆ0 for all x2 R. In (1:16), we replacexyA(x) fory and then getA(x)xyA(x)xˆ0 for allx;y2 R. By the semiprimeness of Rit follows that

A(x)xˆ0 (1:17)

for allx2 R. Left multiplication of (1:16) byxand right multiplication of (1:16) by A(x) lead to xA(x)yxA(x)ˆ0 for all x;y2 R. In view of the semiprimeness ofR, we get

xA(x)ˆ0 for allx2 R. The linearization of (1:17) leads to

A(x)y‡B(x;y)x‡A(y)x‡B(x;y)yˆ0 (1:18)

for all x;y2 R, where B(x;y) denotes m(xy‡yx) m(x)y m(y)x xd(y) yd(x). Replacing xforxin (1:18), we have

A(x)y‡B(x;y)x A(y)x B(x;y)yˆ0 (1:19)

for allx;y2 R. Combining (1:19) with (1:18), we obtain A(x)y‡B(x;y)xˆ0

for allx;y2 R. Right multiplication of the above relation byA(x) leads to A(x)yA(x)ˆ0 for allx;y2 R. In view of the semiprimeness ofR, we get A(x)ˆ0 for allx2 R. This proves thatm(x²)ˆm(x)x‡xd(x) for allx2 R and that m is a generalized Jordan derivation of R with the associated Jordan derivationd. By [27, Theorem 2.6] it follows thatmis a generalized

derivation with the associated derivationd. p

2. Pair of Generalized Derivations on (Semi-)Prime Rings.

LetRbe a ring with the centerZ. A mappingf :R ! Ris said to be skewcentralizing f(x)x‡xf(x)2 Z for allx2 R. The study of skew cen- tralizing mappings was motivated by a well-known theorem of Posner which says that the existence of a nonzero centralizing derivation on a prime ringRimplies thatRis commutative [18]. This theorem has been generalized by many researchers in different ways. One interesting branch of all related works is to consider a common generalization of skew cen- tralizing mappings which involves pair of generalized Jordan derivations on prime rings. In this section, one much more common generalized form of skew centralizing mapping such asm(x^m)xⁿ‡xⁿn(x^m)2 Z will be stu-

died, where m;n are fixed positive integers, m and n are a pair of gen- eralized Jordan derivations on prime ringR.

For the proof of our main result of this section, we need some basic facts. Through out this sectionRalways denotes a (semi-)prime ring andU is theleft Utumi quotient ringofR.U can be characterized as a ring sa- tisfying the following properties:

(a) Ris a subring ofU.

(b) For eachq2 U, there exists a dense left idealI_q ofRsuch that Iqq R.

(c) Ifq2 U andIqˆ0 for some idealIofR, thenqˆ0.

(d) Iff:I! Ris a leftR-module homomorphism from a dense left idealIofRintoR, then there exists an elementq2 Usuch that f(i)ˆiqfor alli2I:

Up to isomorphisms, U is uniquely determined by the above four properties. The center of U is called the extended centroid of R and denoted byC. It is well known thatCis a Von Neumann regular ring. It turns out thatCis a field if and only ifRis a prime ring. The set of all idempotents of C is denoted by E. The element ofE are called central idempotents.

Another related object we have to mention is the generalized differ- ential identities on (semi-)prime rings. A generalized differential poly- nomial overU means a generalized polynomial with coefficients inU and with noncommutative variables involving generalized derivations. A gen- eralized differential identity for some subset ofU is a generalized differ- ential polynomial satisfied by the given subset. Obviously, the definition of a generalized differential polynomial(or identity) is a common general- ization of the definition of a differential polynomial(or identity). Before we state the main result of this section, several useful lemmas are given.

LEMMA2.1. Let n be a fixed positive integer,Rbe a n!-torsion free ring with centerZ. Suppose y1;y2; ;yn2 Rsatisfyly1‡l²y2‡ ‡lⁿyn2 Z forlˆ1;2; ;n. Then yi2 Zfor all i.

The Lemma 1 of [5] has been used many times in section 1 and Lemma 2.1 is actually one generalization of this result. The proof of Lemma 2.1 is completely analogous to the proof of Lemma 1 of [5].

LEMMA2.2 [26, Theorem 2]. LetRbe a semiprime ring and I be a dense ideal ofR. Then any generalized derivationm on I can be uniquely ex- tended to a generalized derivation of U. Furthermore, the extended gen-

eralized derivation m has the form m(x)ˆax‡d(x) for all x2 U, where a2 U and d is a derivation ofU.

Now we are ready to give the first main result of this paper.

THEOREM 2.3. Let m;n be fixed positive integers, R be a non- commutative2(m‡n)!-torsion free prime ring andm;nbe a pair of gen- eralized Jordan derivations onR. Ifm(x^m)xⁿ‡xⁿn(x^m)2 Zfor all x2 R, thenmandnare right(or left)multipliers.

PROOF. Sincemandnare generalized Jordan derivations onR,mandn are generalized derivations onRby [27, Theorem 2.6]. It is well known that RandUsatisfy the same differential identities [16, Theorem 2] and hence also satisfy the same generalized differential identities by Lemma 2.2. This implies that

m(x^m)xⁿ‡xⁿn(x^m)2 C (2:1)

for allx2 U. Note thatUhas the identity elemente. It is easy to see that m(e)‡n(e)2 C:

(2:2)

Substitutingx‡lyforxin (2:1) and applying Lemma 2.1, we get P1(x;y)ˆm(x^{m 1}y‡x^{m 2}yx‡ ‡yx^{m 1})xⁿ

(2:3)

‡m(x^m)(x^{n 1}y‡x^{n 2}yx‡ ‡yx^{n 1})‡(x^{n 1}y‡x^{n 2}yx‡ ‡yx^{n 1})n(x^m)

‡xⁿn(x^{m 1}y‡x^{m 2}yx‡ ‡yx^{m 1})2 C:

for all x;y2 U, where l2Z and Pi(x;y) denotes the sum of terms involving i factors of y in the expansion of m((x‡ly)^m)(x‡ly)ⁿ‡

‡(x‡ly)ⁿn((x‡ly)^m)2 C. Takingxˆeinto (2:3) leads to mm(y)‡nm(e)y‡nyn(e)‡mn(y)2 C:

(2:4)

for all y2 U. It follows from (2:4) and Lemma 2.2 that

(m‡n)m(e)y‡nyn(e)‡mn(e)y‡md(y)‡mg(y)2 C:

(2:5)

for all y2 U, where d and g are the associated derivations of m and n, respectively. Thus

n[m(e)y‡yn(e);y]‡m[d(y)‡g(y);y]ˆ0 (2:6)

for all y2 U. Rewrite (2:6) as follows

([nm(e);y]‡m(d(y)‡g(y)))y y([y;nn(e)]‡m(d(y)‡g(y)))ˆ0 (2:7)

for ally2 U. By [3, Theorem 4.1] we have

n[m(e);y]‡m(d(y)‡g(y))ˆ0 (2:8)

n[y;n(e)]‡m(d(y)‡g(y))ˆ0 (2:9)

for ally2 U. Combining (2:8) with (2:5), we obtain m(m(e)‡n(e))y‡ny(m(e)‡n(e))2 C:

(2:10)

for ally2 U. Note thatm(e)‡n(e)2 CandRis 2(m‡n)!-torsion free. So m(e)ˆ n(e):

(2:11)

By Lemma 2.2, the relation (2:1) can be set

m(e)x^m‡n‡xⁿn(e)x^m‡d(x^m)xⁿ‡xⁿg(x^m)2 C:

(2:12)

Replacingybyxⁿin (2:8) and then right multiplication byx^mgives nm(e)x^n‡m nxⁿm(e)x^mˆ mg(xⁿ)x^m md(xⁿ)x^m for allx2 U. According to the above relation and (2:12), we have

mg(xⁿ)x^m md(xⁿ)x^m‡nd(x^m)xⁿ‡nxⁿg(x^m)2 C (2:13)

for allx2 U. Substitutingx‡lyforxin (2:13) and applying Lemma 2.1 yields

(2:14) mg(x^{n 1}y‡x^{n 2}yx‡ ‡yx^{n 1})x^m

md(x^{n 1}y‡x^{n 2}yx‡ ‡yx^{n 1})x^m mg(xⁿ)(x^{m 1}y‡x^{m 2}yx‡ ‡yx^{m 1}) md(xⁿ)(x^{m 1}y‡x^{m 2}yx‡ ‡yx^{m 1})

‡nd(x^{m 1}y‡x^{m 2}yx‡ ‡yx^{m 1})xⁿ

‡nxⁿg(x^{m 1}y‡x^{m 2}yx‡ ‡yx^{m 1})

‡nd(x^m)(x^{n 1}y‡x^{n 2}yx‡ ‡yx^{n 1})

‡n(x^{n 1}y‡x^{n 2}yx‡ ‡yx^{n 1})g(x^m)2 C for allx;y2 U. Takingyˆeinto (2:14), we obtain

mng(x^{n 1})x^m mnd(x^{n 1})x^m m²g(xⁿ)x^{m 1} m²d(xⁿ)x^{m 1} mnd(x^{m 1})xⁿ‡mnxⁿg(x^{m 1})‡n²d(x^m)x^{n 1}‡n²x^{n 1}g(x^m)2 C (2:15)

for allx2 U. Continuing the process between (2:13)-(2:15) and applying the fact thatUis torsion free we ultimately get

(n‡m)g(x)x‡(n m)d(x)x (n m)xd(x)‡(n‡m)xg(x)2 C for allx2 U. This is

[(n m)d(x) (n‡m)g(x);x]2 C for allx2 U. By [18, Lemma 3] it follows that

(n‡m)g(x)ˆ(n m)d(x) (2:16)

for allx2 U.

CASE1. Ifmˆn, then (n‡m)g(x)ˆ0 for allx2 U. Due to torsion free fact of the ringU,g(x)ˆ0. Thus (2:8) becomes as follows

[m(e);x]‡d(x)ˆ0

for allx2 U. This leads tom(x)ˆm(e)x‡d(x)ˆxm(e), which meansmis a right multiplier.

CASE2. Ifm6ˆn, by (2:8), (2:16) and a little computation, we have (m‡n)[m(e);x]‡2md(x)ˆ0

(2:17)

(m n)[m(e);x] 2mg(x)ˆ0 (2:18)

for allx2 U. Replacingx^mforxin (2:17) and (2:18) leads to (m‡n)[m(e);x^m]‡2md(x^m)ˆ0 (2:19)

(m n)[m(e);x^m] 2mg(x^m)ˆ0 (2:20)

for allx2 U. Combining (2:11) with (2:12) we get

2mm(e)x^m‡n 2mxⁿm(e)x^m‡2md(x^m)xⁿ‡2mxⁿg(x^m)2 C:

(2:21)

By (2:19), (2:20) and (2:21) we obtain

(2mm(e)x^m (m‡n)[m(e);x^m])xⁿ xⁿ(2mm(e)x^m (m n)[m(e);x^m])2 C for allx2 U. That is

(n m)[x^n‡m;m(e)] (m‡n)x^m[x^{n m};m(e)]x^m2 C (2:22)

for allx2 U. Let us setdm(e)(x)ˆ[x;m(e)]. Then (2:22) can be rewritten (n m)d_m(e)(x^m‡n) (m‡n)x^md_m(e)(x^{n m})x^m2 C

(2:23)

for all x2 U. Substitutingx‡lyforxin (2:23) and applying Lemma 2.1 yields

(n m)d_m(e)(x^{n‡m 1}y‡x^{n‡m 2}yx‡ ‡yx^{n‡m 1}) (2:24)

(m‡n)(x^{m 1}y‡x^{m 2}yx ‡yx^{m 1})d_m(e)(x^{n m})x^m (m‡n)x^mdm(e)(x^{n m})(x^{m 1}y‡x^{m 2}yx ‡yx^{m 1})

(m‡n)x^md_m(e)(x^{n m 1}y‡x^{n m 2}yx‡ ‡yx^{n m 1})x^m2 C for allx2 U. Takingyˆeinto (2:24) leads to

(n m)(m‡n)d_m(e)(x^{n‡m 1}) m(m‡n)x^{m 1}d_m(e)(x^{n m})x^m m(m‡n)x^mdm(e)(x^{n m})x^{m 1} (n m)(m‡n)x^mdm(e)(x^{n m 1})x^m2 C for allx2 U. SinceU is a 2(m‡n)!-torsion free ring,

(n m)d_m(e)(x^{n‡m 1}) mx^{m 1}d_m(e)(x^{n m})x^m (2:25)

mx^mdm(e)(x^{n m})x^{m 1} (n m)x^mdm(e)(x^{n m 1})x^m2 C:

for allx2 U. Comparing (2:23) with (2:25), we have by induction dm(e)(x³) 3xdm(e)(x)x2 C

for allx2 U. That is

[[d_m(e)(x);x];x]2 C for allx2 U. By [15, Theorem 1] it follows that

dm(e)(x)ˆ0

for allx2 U. This implies thatm(e)2 C. In view of (2:17) and (2:18), we have dˆ0 andgˆ0. This shows thatmandnare right (or left) multipliers. p

As a direct corollary of Theorem 2.3 we immediately get

THEOREM 2.4. Let m;n be fixed positive integers, R be a non- commutative2(m‡n)!-torsion free prime ring and d;g be a pair of Jordan derivations on R. If d(x^m)xⁿ‡xⁿg(x^m)2 Z for all x2 R, then dˆ0and gˆ0.

Using the orthogonal completeness method [2] we can extend Theorem 2.4 to the case of semiprime rings.

THEOREM 2.5. Let m;n be a fixed positive integer, R be a non- commutative2(m‡n)!-torsion free semiprime ring and d;g be a pair of Jordan derivations on R. If d(x^m)xⁿ‡xⁿg(x^m)2 Z for all x2 R, then d and g both mapRintoZ.

PROOF. By Cusack's Theorem [6] we know thatdandgare derivations onR. LetBbe the complete Boolean algebra ofE. We choose a maximal idealMofB. According to [2],MUis a prime ideal ofU, which is invariant under any derivation ofU. It is well known that the pair of derivationsd;g onRcan be uniquely extended to be a pair of derivations onU. Letd;gbe the canonical pair of derivations on U ˆ U=MU induced by d;g, respec- tively. The assumption implies that

[d(x^m)xⁿ‡xⁿg(x^m);z]ˆ0

for allx;z2 R. It follows from [16, Theorem 2] thatRandU satisfy the same differential identities. Thus

[d(x^m)xⁿ‡xⁿg(x^m);z]ˆ0 for allx;z2 U. Furthermore,

[d(x^m)xⁿ‡xⁿg(x^m);z]ˆ0;

for allx;z2 U. By Theorem 2.4 we know that eitherd(x)ˆ0 andg(x)ˆ0 or [U;U]ˆ0. In any case, we have both

d(U)[U;U]2 MU and

g(U)[U;U]2 MU for allM. Note that T

fMUjMis any maximal ideal of Bg ˆ0. Hence d(U)[U;U]ˆ0 and g(U)[U;U]ˆ0. In particular, d(R)[R;R]ˆ0 and g(R)[R;R]ˆ0. These imply that

0ˆd(R)[R²;R]ˆd(R)R[R;R]‡d(R)[R;R]R ˆd(R)R[R;R]

and

0ˆg(R)[R²;R]ˆg(R)R[R;R]‡g(R)[R;R]R ˆg(R)R[R;R]:

Therefore [R;d(R)]R[R;d(R)]ˆ0 and [R;g(R)]R[R;g(R)]ˆ0. By semi- primeness ofRwe obtain that [R;d(R)]ˆ0 and [R;g(R)]ˆ0. These show

thatd(R)2 Zandg(R)2 Z. p

THEOREM 2.6. Let m;n;p;q be fixed non-negative integers, R be a noncommutative 2(m‡n)!-torsion free prime ring and d;g be a pair of Jordan derivations on R. If x^md(x)xⁿ‡x^pg(x)x^q2 Z for all x2 R, then dˆ0and gˆ0, where m6ˆp or n6ˆq.

PROOF. Sincedandgare Jordan derivations onR,dandgalso are derivations on Rby Herstein's Theorem [12]. It is well known that R and U satisfy the same differential identities [16, Theorem 2]. This im- plies that

x^md(x)xⁿ‡x^pg(x)x^q2 C (2:26)

for allx2 U. Substitutingx‡lyforxin (2:26) and applying Lemma 2.1 yields

P1(x;y)ˆ(x^{m 1}y‡x^{m 2}yx‡ ‡yx^{m 1})d(x)xⁿ‡x^md(y)xⁿ (2:27)

‡x^md(x)(x^{n 1}y‡x^{n 2}yx‡ ‡yx^{n 1})

‡(x^{p 1}y‡x^{p 2}yx‡ ‡yx^{p 1})g(x)x^q

‡x^pg(y)x^q‡x^pg(x)(x^{q 1}y‡x^{q 2}yx‡ ‡yx^{q 1})2 C for allx;y2 U, whereP_i(x;y) denotes the sum of terms involvingifactors ofy. Note thatUhas the identity elemente. Takingxˆeinto (2:27) we get

d(y)‡g(y)2 C

for ally2 U. By the well known Posner's theorem [18] it follows that d(y)‡g(y)ˆ0

(2:28)

for ally2 U. Using a similar computational way to (2:27), we also have P2(e;y)ˆmyd(y)‡nd(y)y‡pyg(y)‡qg(y)y2 C

(2:29)

for ally2 U. Comparing (2:29) with (2:28) we get (n q)d(y)y‡(m p)yd(y)2 C (2:30)

for ally2 U. Ifn qˆm p6ˆ0, thendˆ0 by the torsion free fact ofU.

Ifn q6ˆm p, linearizing the identity (2:30) yields

(n q)d(x)y‡(n q)d(y)x‡(m p)xd(y)‡(m p)yd(x)2 C (2:31)

for allx;y2 U. Takingyˆeinto (2:31) leads to (n q)d(x)‡(m p)d(x)2 C (2:32)

for allx2 U. Ifn qˆ m‡p, then (2:30) becomes as follows [d(y);y]2 C

for ally2 U. By Posner' theorem we know thatdˆ0. Ifn q6ˆ m‡p, then (2:32) also makedˆ0. In any case ofm6ˆporn6ˆq,dˆ0 always

holds. Sogˆ0 by (2:28). p

Similarly, we also adopt the orthogonal completeness method [2] to prove the semiprime version corresponding to Theorem 2.6. Its proof is completely analogous to that of Theorem 2.5 and is not presented here. We only state it as

THEOREM 2.7. Let m;n;p;q be fixed non-negative integers, R be a noncommutative2(m‡n)!-torsion free semiprime ring and d;g be a pair of Jordan derivations onR. If x^md(x)xⁿ‡x^pg(x)x^q 2 Zfor all x2 R, then d and g both mapRintoZ, where m6ˆp or n6ˆq.

3. Pairs of Derivations on Banach Algebras.

In view of the above algebraic results, we will focus several range in- clusion problems involved pair of derivations on Banach algebras in this section. Aalways denotes aBanach algebra which is a complex normed algebra and its underlying vector space is a Banach space. TheJacobson radicalofAis the intersection of all primitive ideals ofAand is denoted by rad(A). Thenil radicalofAis the intersection of all prime ideals ofAand is denoted bynil(A). LetI be any closed ideal of the Banach algebraA.

ThenQ_I denotes the canonical quotient map fromAontoA=I. Moreover, we assume that all mappings on Banach algebraAare linear mappings in the whole paper.

At present, most of results in this field deal with one derivation, while pair of derivations on Banach algebras was more less considered. On the other hand, many results are proved under the assumption of continuity of derivations. Our several results are proved without assuming the con- tinuity of derivations. Let us begin this section with the following lemma.

LEMMA3.1 [23, Lemma 1.2]. Let d be a derivation on Banach algebraA andJ be a primitive ideal ofA. If there exists a real constant k>0such thatkQ_Jdⁿk kⁿfor all n2N, then d(J) J.

Now we are in position to prove the main result of this section.

THEOREM3.2. LetAbe a noncommutative Banach algebra and d;g be a pair of derivations on A. If d(x^m)xⁿ‡xⁿg(x^m)2 ZA for all x2 Aand fixed positive integers m;n, then d(A)rad(A)and g(A)rad(A).

PROOF. LetJ be any primitive ideal ofA. By Zorn lemma, there exists a minimal prime idealPofAcontained inJ. Thend(P) Pandg(P) P, by [17, Lemma]. IfP is closed, thendandginduce the derivations on the Banach algebraA=Pas follows

~d(~x)ˆd(x)‡ P; ~g(~x)ˆg(x)‡ P

for all ~x2 A=P andx2 A. WhenA=P is commutative, both~d(A=P) and

~

g(A=P) are contained in the Jacobson radical ofA=Pby [22, Theorem 4.4].

WhenA=Pis noncommutative, the assumption yields [~d(~x^m)~xⁿ‡~xⁿ~g(~x^m);~z]ˆ~0

for all~x;~z2 A=Pandx;z2 A. By the primeness ofA=Pand Theorem 2.4, it follows that~dˆ~0 and~gˆ~0 onA=P. In any case, we get bothd(A) Jand g(A) J. IfPis not closed, thenS(d) Pby [6, Lemma 2.3], whereS(d) is the separating space of linear operator d. By [20, Lemma 1.3], we have S(QP^d)ˆQP^d(S(d))ˆ0, whenceQP^dis continuous onA. This implies that QP^d(P)^ ˆ0 onA=Pand henced(P)^ P. Thus^ dinduces a derivation on the Banach algebraA=P^ as follows

~d(~x)ˆd(x)‡P^

for all~x2 A=P^ andx2 A. Thus we can define the following map jd~ⁿQP^ :A ! A=P ! A=^ P ! A=J^

through j~dⁿQ_P^(x)ˆQ_Jdⁿ(x) for all x2 Aandn2N, wherejis the ca- nonical inclusion map fromA=P^ontoA=J andjindeed exists sinceP J^ . By [20, Lemma 1.4], we claim thatd~is continuous onA=P^ and hence that kQ_Jdⁿk kdk~ ⁿ for alln2N. By Lemma 3.1, we obtaind(J) J. Using the same argument withg, we also obtaing(J) J. Then the derivationsd andginduce the derivations on the Banach algebraA=J as follows

~d(~x)ˆd(x)‡ J; ~g(~x)ˆg(x)‡ J

for all~x2 A=Jandx2 A. The remainder follows by a similar argument to the case when P is closed since the primitive algebra A=J is prime.

Therefore we conclude that d(A) J and g(A) J. It follows that

d(A) J andg(A) J for every primitive idealJ, that is,d(A)rad(A) andg(A)rad(A). This completes the proof of the theorem. p

As a consequence of the above theorem, we get

COROLLARY 3.3. Let A be a noncommutative semisimple Banach algebra and d;g be a pair of derivations on A. If d(x^m)xⁿ‡

‡xⁿg(x^m)2 ZA for all x2 Aand fixed positive integers m;n, then dˆ0 and gˆ0.

REMARK3.4. Simulating the preceding proof ways which just appear in this section, some similar results can be obtained. In fact, Theorem 3.2 and Corollary 3.3 still hold if the conditiond(x^m)xⁿ‡xⁿg(x^m)2 Z_Afor all x2 A is replaced by the condition d(x^m)xⁿ‡xⁿg(x^m)2nil(A) for all x2 A. Here the assumption of continuity of derivations is not necessary.

We only state the final results without detailed proof.

THEOREM3.5. LetAbe a noncommutative Banach algebra and d;g be a pair of derivations onA. If d(x^m)xⁿ‡xⁿg(x^m)2nil(A)for all x2 Aand fixed positive integers m;n, then d(A)rad(A)and g(A)rad(A).

COROLLARY 3.6. Let A be a noncommutative semisimple Banach algebra and d, g be a pair of derivations on A. If d(x^m)xⁿ‡

‡xⁿg(x^m)2nil(A) for all x2 A and fixed positive integers m;n, then dˆ0and gˆ0.

Let us consider the case of a pair of Jordan derivations on Banach al- gebras.

THEOREM3.7. LetAbe a noncommutative Banach algebra and d;g be a pair of continuous Jordan derivations on A. If d(x^m)xⁿ‡xⁿg(x^m)2 2rad(A)for all x2 Aand fixed positive integers m;n, then d(A)rad(A) and g(A)rad(A).

PROOF. LetPbe any primitive ideal ofA. Sincedandgare continuous, d(P) Pandg(P) Pby [19, Lemma 3.2]. Thendandgcan be induced to the Jordan derivations on the Banach algebraA=Pas follows

d(~~x)ˆd(x)‡ P; ~g(~x)ˆg(x)‡ P

for all~x2 A=Pandx2 A. SincePis a primitive ideal, the quotient algebra

A=P is prime and semisimple. Thus bothd~ and~g are derivations by [12, Theorem 3.1]. It has been proved that every derivation on a semisimple Banach algebra is continuous [13, Remark 4.3]. Combing this result with the well known Singer-Wermer theorem, we conclude that there are no nonzero derivations on a commutative semisimple Banach algebra. Hence we haved~ˆ0 and~gˆ0 whenA=Pis commutative. It remains to show that d~ˆ0 and~gˆ0 in the case whenA=Pis noncommutative. The assumption of the theorem yields

~d(~x^m)~xⁿ‡~xⁿg(~x^m)ˆ~0

for all~x2 A=Pandx2 A. It follows from Theorem 2.4 thatd~ˆ0 and~gˆ0.

In any case both~dˆ0 and~gˆ0. This implies thatd(A) Pandg(A) P.

SincePis arbitrary,d(A)rad(A) andg(A)rad(A). This completes the

proof of the theorem. p

The following corollary is a special case of Theorem 3.7.

COROLLARY 3.8. Let A be a noncommutative semisimple Banach algebra and d;g be a pair of Jordan derivations onA. If

d(x^m)xⁿ‡xⁿg(x^m)2rad(A)

for all x2 Aand fixed positive integers m;n, then dˆ0and gˆ0.

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Manoscritto pervenuto in redazione il 2 dicembre 2008.