Lie ideals and Jordan triple derivations in rings

Lie Ideals and Jordan Triple Derivations in Rings

MOTOSHIHONGAN(*) - NADEEM URREHMAN(**) - RADWANMOHAMMED

AL-OMARY(**)

ABSTRACT- In this paper we prove that on a2-torsion free semiprime ringRevery Jordan triple (resp. generalized Jordan triple) derivation on Lie idealL is a derivation onL(resp. generalized derivation onL) .

1. Introduction.

This paper is motivated by the work of Jing and Lu [13]. Throughout the present paper, Rwill denote an associative ring with centerZ(R). A ringRis said to be 2-torsion free, if whenever 2xˆ0 withx2R, implies xˆ0. Recall that Ris prime if for any a;b2R,aRbˆ f0g implies that aˆ0 or bˆ0, and R is semiprime if for all a2R, aRaˆ f0g implies aˆ0. An additive subgroup L of R is said to be a Lie ideal of R if [L;R]L. A Lie idealLis said to be square-closed ifa²2Lfor alla2L.

An additive mappingd:R !Ris called a derivation (resp. a Jordan de- rivation) if (xy)^dˆx^dy‡xy^d(resp. (x²)^dˆx^dx‡xx^d) holds for allx;y2R.

A famous result due to Herstein [11, Theorem 3.3] states that a Jordan derivation of a prime ring of characteristic not equal to 2 must be a deri- vation. A brief proof of Herstein's result can also be found in [8]. This result was extended to 2-torsion free semiprime rings by Cusack [10] and sub- sequently, by Bresar [7] .

Following [6], an additive mappingd:R !Ris called a Jordan triple derivation if (xyx)^dˆx^dyx‡xy^dx‡xyx^d holds for allx;y2R. One can

(*) Indirizzo dell'A.: Department of Mathematics, Tsuyama College of Technology, 624-1 Numa, Tsuyama, Okayama (Japan).

E-mail: hongan@mx32.tiki.ne.jp

(**) Indirizzo dell'A.: Department of Mathematics, Aligarh Muslim University, Aligarh 202002 (India).

E-mail: rehman100@gmail.com, radwan959@yahoo.com

This research is supported by UGC, India, Grant No. 36-8/2008(SR).

easily prove that any Jordan derivation on an arbitrary 2-torsion free ring is a Jordan triple derivation (see for example [8] where further references can be found). Bresar has proved the following result.

THEOREM1.1 (6, Theorem 4.3). Let R be a2-torsion free semiprime ring and letd:R!R be a Jordan triple derivation. In this cased is a derivation.

An additive mappingF:R !Ris said to be a generalized derivation (resp. a generalized Jordan derivation) on R if there exists a derivation d:R !Rsuch that (xy)^Fˆx^Fy‡xy^d(resp. (x²)^F ˆx^Fx‡xx^d) holds for all x;y2R. An additive mapping F:R !R is said to be a generalized Jordan triple derivation on R if there exists a Jordan triple derivation d:R !Rsuch that (xyx)^F ˆx^Fyx‡xy^dx‡xyx^d holds for allx;y2R.

Motivated by the above result Jing and Lu have proved the following generalization of Theorem 1.1.

THEOREM1.2 (13, Theorem 3.5). Let R be a2-torsion free prime ring, then every generalized Jordan triple derivation on R is a generalized de- rivation.

Very recently, Vukman [18] extended the result mentioned above for 2- torsion free semiprime rings. In the present paper, our objective is to generalize Theorem 1.1 and 1.2 on Lie ideal ofR.

2. Jordan Triple Derivations.

It is obvious to see that evey derivation is a Jordan triple derivation, but the converse need not to be true in general. Motivated by the Theorem 1.1, in the present section, it is shown that on a 2-torsion free semiprime ringR every Jordan triple derivation on Lie ideal Lis a derivation on L. More precisely, we prove the following:

THEOREM2.1. Let R be a2-torsion free semiprime ring and L6Z(R) be a square-closed Lie ideal of R. If an additive mappingd:R !R sa- tisfies

(aba)^d ˆa^dba‡ab^da‡aba^d for all a;b2L and L^dL, thendis a derivation on L.

To facilitate our discussion, we shall begin with the following lemmas:

LEMMA2.1 (4, Lemma 4). Let L6Z(R)be a Lie ideal of a2-torsion free prime ring R and a;b2R. If aLbˆ f0g, then aˆ0or bˆ0.

LEMMA2.2 (17, Lemma 2.4). Let R be a2-torsion free semiprime ring, L be a Lie ideal of R such that L6Z(R)and let a2L. If aLaˆ f0g, then a²ˆ0and there exisits a nonzero ideal KˆR[L;L]R of R generated by [L;L]such that[K;R]L and KaˆaKˆ f0g.

COROLLARY2.1. LetRbe a2-torsion free semiprime ring,Lbe a Lie ideal ofRsuch thatL6Z(R)and leta; b2L.

(1)If aLaˆ f0g, then aˆ0.

(2)If aLˆ f0g(or Laˆ f0g), then aˆ0.

(3)If L is square-closed, and aLbˆ f0g, then abˆ0and baˆ0.

PROOF. (1) SinceKaˆR[L;L]Raˆ f0ganda²ˆ0 by Lemma 2.2, we have 0ˆ[[a;x];a]yaˆaxayafor allx;y2R, and so, we haveaxayaxaˆ0.

Since R is semiprime, we get axaˆ0 for allx2R. Moreover, by semi- primeness ofR, we getaˆ0.

(2) It is clear by (1).

(3) IfaLbˆ f0g, then we havebaLbaˆ f0gobviously, and so we have baˆ0 by (1). Moreover, sinceabLabaLbˆ f0g, we getabˆ0.

LEMMA 2.3 (17, Lemma 2.5). Let R be a2-torsion free ring, L be a Lie ideal of R and let a;b2L. If aub‡buaˆ0for all u2L, then aubLaubˆ f0g.

LEMMA2.4 (17, Lemma 2.7). Let G₁;G₂; ;G_nbe additive groups, R a 2-torsion free semiprime ring and L6Z(R)be a Lie ideal of R. Suppose that mappings S:G1G2 Gn !R and T:G1G2 Gn !R are addtitive in each argument. If S(a1;a2; ;an)xT(a1;a2; ;an)ˆ0for all x2L, a_i2G_i iˆ1;2; n, then S(a1;a2; ;an)xT(b1;b2; ;bn)ˆ0 for all x2L, a_i;b_i2G_iiˆ1;2; n.

LEMMA2.5. Letdbe a Jordan triple derivation and L be a Lie ideal of R. For arbitrary a;b;c2L, we have

(abc‡cba)^dˆa^dbc‡ab^dc‡abc^d‡c^dba‡cb^da‡cba^d:

PROOF. We have

(aba)^dˆa^dba‡ab^da‡aba^d for alla;b2L:

(2:1)

We computeWˆ f(a‡c)b(a‡c)g^din two different ways. On one hand, we find thatWˆ(a‡c)^db(a‡c)‡(a‡c)b^d(a‡c)‡(a‡c)b(a‡c)^d, and on the other handWˆ(aba)^d‡(abc‡cba)^d‡(cbc)^d. Comparing two ex- pressions, we obtain the required result.

REMARK 2.1. It is easy to see that every Jordan derivation of a 2- torsion free ring satisfies (2.1) ( see [1] for reference).

For the purpose of this section, we shall write; D(abc)ˆ(abc)^d a^dbc ab^dc abc^d; and L(abc)ˆabc cba. We list a few elementary properties ofdandL:

(i)D(abc)‡D(cba)ˆ0

(ii)D((a‡b)cd)ˆD(acd)‡D(bcd) andL((a‡b)cd)ˆL(acd)‡L(bcd) (iii)D(a(b‡c)d)ˆD(abd)‡D(acd) andL(a(b‡c)d)ˆL(abd)‡L(acd) (iv)D(ab(c‡d))ˆD(abc)‡D(abd) andL(ab(c‡d))ˆL(abc)‡L(abd) PROPOSITION 2.2. Let R be a 2-torsion free semiprime ring and L6Z(R) be a square closed Lie ideal of R. If D(abc)ˆ0 holds for all a; b; c2L, thendis a derivation of L.

PROOF. Let assumeD(abc)ˆ0 for alla;b;c2L, that is, (abc)^dˆa^dbc‡ab^dc‡abc^d:

LetMˆabxab. We have

M^dˆ fa(bxa)bg^dˆ(a)^d(bxa)b‡ab^dxab‡abx^dab

‡(abx)a^db‡(abxa)b^d for allx;a;b2L:

(2:2)

On the other hand,

M^dˆ f(ab)x(ab)g^dˆ(ab)^d(xab)‡(ab)x^d(ab)‡(abx)(ab)^d: (2:3)

Comparing (2.3) with (2.2) we get

f(ab)^d a^db ab^dg(xab)‡(abx)f(ab)^d a^db ab^dg ˆ0

that is,a^bxab‡abxa^bˆ0, wherea^bstands for (ab)^d a^db ab^d. Thus, by Lemma 2.3 we find that (a^bxab)y(a^bxab)ˆ0 for alla;b;x;y2L. Now, by

Corollary 2.1, we obtaina^bxabˆ0 for alla;b;x2L. Now by Lemma 2.4, we obtaina^bxcdˆ0 for alla;b;c;d;x2L. Hence, by using Corollary 2.1, we geta^bˆ0 for alla;b2Lthat is,dis a derivation onL.

LEMMA2.6. Let L be a Lie ideal of R. For any a;b;c;x2L, we have D(abc)xL(abc)‡L(abc)xD(abc)ˆ0:

PROOF. For any a;b;c;x2L, suppose that Nˆabcxcba‡cbaxabc.

Now we find

N^d ˆ fa(bcxcb)a‡c(baxab)cg^dˆ fa(bcxcb)ag^d‡ f(c(baxab)cg^d

ˆ a^dbcxcba‡ab^dcxcba‡abc^dxcba

‡abcx^dcba‡abcxc^dba‡abcxcb^da

‡abcxcba^d‡c^dbaxabc‡cb^daxabc

‡cba^dxabc‡cbax^dabc‡cbaxa^dbc

‡cbaxab^dc‡cbaxabc^d: On the other hand, we have

N^d ˆ f(abc)x(cba)‡(cba)x(abc)g^d

ˆ (abc)^dx(cba)‡(abc)x^d(cba)‡(abcx)(cba)^d

‡(cba)^d(xabc)‡(cba)x^d(abc)‡(cbax)(abc)^d: By comparing last two expressions, we get

D(cba)(xcba)‡D(cba)(xabc)‡(abcx)D(cba) (cbax)D(cba)ˆ0:

This implies thatD(abc)xL(abc)‡L(abc)xD(abc)ˆ0 for alla;b;c2L.

LEMMA2.7. Let R be a semiprime ring and L6Z(R)be a square closed Lie ideal of R. ThenD(abc)xL(rst)ˆ0holds for all a;b;c;r;s;t;x2L:

PROOF. By Lemma 2.6 we haveD(abc)xL(abc)‡L(abc)xD(abc)ˆ0 for all a;b;c;x2L. Thus, we get D(abc)xL(abc)LD(abc)xL(abc)ˆ f0g by Lemma 2.3, and hence we obtainD(abc)xL(abc)ˆ0, for alla;b;c;x2Lby Corollary 2.1. Now, we find thatD(abc)xL(rst)ˆ0, for alla;b;c;r;s;t;x2L by Lemma 2.4.

LEMMA2.8. Let R be a2-torsion free semiprime ring and L be a square closed Lie ideal of R. IfL(abc)ˆ0for all a;b;c2L, then LZ(R).

PROOF. Assume thatL6Z(R). We haveL(abc)ˆ0 for alla;b;c2L, that is, abcˆcba. Replacing b by 2tb, we get 2atbcˆ2ctba for all

a;b;c;t2L. Again replacingtby 2twand using the fact thatRis 2-torsion free, we getatwbcˆctwbaand hencea(twb)cˆa(bwt)cˆa(bc)twˆawtbc.

Thus, we find thata[t;w]bcˆ0 for alla;b;c;t;w2L. By Corollaray 2.1, we get [t;w]ˆ0 for allt;w2L, that is,Lis a commutative Lie ideal ofR. And so, we have [a;[a;t]]ˆ0 for allt2Rand hence by Sublemma on page 5 of [11],a2Z(R). HenceLZ(R), a contradiction. This completes the proof of the Lemma.

PROOF OFTHEOREM 2.1. Supposed:R !L is a Jordan triple deri- vation onL. Our goal will be to show that dis a derivation of associative triple systems. We know thatD(abc)ˆ D(cba). Hence

2D(abc)xD(abc) ˆD(abc)xfD(abc) D(cba)g

ˆD(abc)xfL(abc)^d‡L(c^dba)

‡L(cb^da)‡L(cba^d)g:

By Lemma 2.7, the above relation reduces to

2D(abc)xD(abc)ˆD(abc)xL(abc)^d (2:4)

Similarly we obtain

2D(abc)xD(abc)ˆL(abc)^dxD(abc) (2:5)

Now we have

0 ˆ fD(abc)xL(abc)‡L(abc)xD(abc)g^d

ˆ D(abc)^dxL(abc)‡D(abc)x^dL(abc)

‡D(abc)xL(abc)^d‡L(abc)^dxD(abc)

‡L(abc)x^dD(abc)‡L(abc)xD(abc)^d and according to Lemma 2.6 and (2.4), and (2.5), we get

0ˆ4D(abc)xD(abc)‡D(abc)^dxL(abc)‡L(abc)xD(abc)^d:

We multiply the relation above from left byD(abc)xD(abc)yfor alla,b,c,x, y2L and by Lemma 2.7 we obtain 4D(abc)xD(abc)yD(abc)xD(abc)ˆ0 for all a;b;c;x;y2L. Since R is a 2-torsion free, it follows that D(abc)xD(abc)yD(abc)xD(abc)ˆ0, that is,D(abc)xD(abc)LD(abc)xD(abc)ˆ f0g.

By Corollary 2.1, we find thatD(abc)xD(abc)ˆ0 and again using Corollary 2.1, we find thatD(abc)ˆ0 for alla;b;c2Land hence by Proposion 2.1, we get the required result.

THEOREM2.2. Let R be a2-torsion free prime ring and L6Z(R)be a nonzero square-closed Lie ideal of R. If an additive mappingd:R !L satisfies that

(aba)^dˆa^dba‡ab^da‡aba^d for all a;b2L;

thendis a derivation on L.

PROOF. By Lemma 2.7, we have D(abc)xL(rst)ˆ0 for all a;b;c, r;s;t;x2L, that is, D(abc)LL(rst)ˆ f0g. Now, either D(abc)ˆ0 or L(rst)ˆ0 by Lemma 2.1. If L(rst)ˆ0 for all r;s;t2L, then we get LZ(R) by Lemma 2.8, a contradiction. On the other hand if D(abc)ˆ0 for all a;b;c2L, then we get the required result by Pro- position 2.1.

3. Generalized Jordan Triple Derivations.

An additive mappingm:R!Ris called a Jordan triple left centralizer onLif (aba)^mˆa^mbafor alla;b2L, and called a Jordan left centralizer on Lif (a²)^mˆa^mafor alla2L.

THEOREM3.1. Let R be a2-torsion free semiprime ring and L6Z(R) be a square-closed Lie ideal. Ifm:R!R is a Jordan triple left centralizer on L, thenmis a Jordan left centralizer on L.

PROOF. By the hypothesis, we have

(aba)^mˆa^mba for alla;b2L:

(3:1)

Replacingabya‡cin (3.1), we find that

f(a‡c)b(a‡c)g^mˆa^mba‡c^mba‡a^mbc‡c^mbcfor alla;b;c2L:

On the other hand, we obtain

f(a‡c)b(a‡c)g^mˆ(abc‡cba)^m‡a^mbc‡c^mba for alla;b;c2L:

Combining last two expressions, we get

(abc‡cba)^mˆa^mbc‡c^mba for alla;b;c2L:

(3:2)

Now replacingcbya²in (3.2), we get

(aba²‡a²ba)^mˆa^mba²‡(a²)^mba:

(3:3)

Again replacingbbyab‡bain (3.1), we get (aba²‡a²ba)^mˆa^maba‡a^mba²: (3:4)

Combining (3.3) and (3.4), we obtain

f(a²)^m a^magbaˆ0 for alla;b2L:

By settingV^aˆ(a²)^m a^ma, we have

V^abaˆ0 for alla;b2L:

(3:5)

Thus, by Corollary 2.1, we obtain

V^aaˆ0ˆaV^a for alla2L:

(3:6)

Linearizing (3.6), we get

V^a‡ba‡V^a‡bbˆ0 for alla;b2L:

(3:7)

Now, we compute V^a‡bˆ f(ab‡ba)^m a^mb b^mag ‡(a²)^m a^ma‡

‡(b²)^m b^mband hence we have

V^a‡bˆF(a;b)‡a^m‡b^m for alla;b2L;

(3:8)

whereF(a;b)ˆ(ab‡ba)^m a^mb b^ma. Thus, in view of (3.8), the expres- sion (3.7) implies that

V^ab‡F(a;b)a‡V^ba‡F(a;b)bˆ0:

(3:9)

Again, replacingaby ain (3.9), we get

V^ab‡F(a;b)a V^ba F(a;b)bˆ0:

(3:10)

Adding (3.9) with (3.10), we find that

V^ab‡F(a;b)aˆ0:

(3:11)

On right multiplication of (3.11) byV^a, we obtain 0ˆV^abV^a‡F(a;b)aV^aˆV^abV^a:

Thus, we obtainV^aˆ0 for alla2Lby Corollary 2.1. This gives thatmis a Jordan left centralizer.

THEOREM3.2. Let R be a2-torsion free semiprime ring and L6Z(R) be a square-closed Lie ideal. If F:R!R is a generalized Jordan triple derivation on L with a Jordan triple derivationdsuch that L^dL, then F is a generalized Jordan derivation on L.

PROOF. SinceFis a generalized Jordan triple derivation onL. There- fore we have

(aba)^Fˆa^Fba‡ab^da‡aba^d for alla;b2L:

(3:12)

In (3.12), we takedas a Jordan triple derivation onL. SinceRis a 2-torsion free semiprime ring, so in view of Theorem 2.1dis a derivation onL. Now we writeGˆF d. Then we have

(aba)^G ˆ (aba)^{F d}

ˆ (aba)^F (aba)^d

ˆ (a^F a^d)baˆa^{F d}baˆa^Gba for alla;b2L:

And so, we have (aba)^G ˆa^Gbafor alla;b2L. In otherwords,Gis a Jordan triple left centralizer onL. SinceRis a 2-torsion free semiprime ring, one can conclude thatGis a Jordan left centralizer by Theorem 3.1. HenceFis of the formFˆG‡d, wheredis a derivation andGis a Jordan left cen- tralizer onL. HenceFis a generalized Jordan derivation onL.

THEOREM3.3. Let R be a2-torsion free prime ring and L6Z(R)be a square-closed Lie ideal. If F:R!R is a generalized Jordan triple deri- vation on L, then F is a generalized derivation on L.

PROOF. By Theorem 3.2 and Theorem of [2] we get the required result.

In conclusion, it is tempting to conjecture as follows:

CONJECTURE 3.1. Let R be a 2-torsion free semiprime ring and L6Z(R)be a Lie ideal of R. If F:R!R is a generalized Jordan triple derivation on L, then F is a generalized derivation on L.

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