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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2009, Article IDΒ 582181, 8 pages
http://dx.doi.org/10.1155/2009/582181
Research Article

Vanishing Power Values of Commutators with Derivations on Prime Rings

Department of Mathematics, Belda College, Belda, Paschim Medinipur 721424, India

Received 30 August 2009; Accepted 14 December 2009

Academic Editor: HowardΒ Bell

Copyright Β© 2009 Basudeb Dhara. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let 𝑅 be a prime ring of char 𝑅≠2, 𝑑 a nonzero derivation of 𝑅 and 𝜌 a nonzero right ideal of 𝑅 such that [[𝑑(π‘₯),π‘₯]𝑛,[𝑦,𝑑(𝑦)]π‘š]𝑑=0 for all π‘₯,π‘¦βˆˆπœŒ, where 𝑛β‰₯0, π‘šβ‰₯0, 𝑑β‰₯1 are fixed integers. If [𝜌,𝜌]πœŒβ‰ 0, then 𝑑(𝜌)𝜌=0.

1. Introduction

Throughout this paper, unless specifically stated, 𝑅 always denotes a prime ring with center 𝑍(𝑅) and extended centroid 𝐢, 𝑄 the Martindale quotients ring. Let 𝑛 be a positive integer. For given π‘Ž,π‘βˆˆπ‘…, let [π‘Ž,𝑏]0=π‘Ž and let [π‘Ž,𝑏]1 be the usual commutator π‘Žπ‘βˆ’π‘π‘Ž, and inductively for 𝑛>1, [π‘Ž,𝑏]𝑛=[[π‘Ž,𝑏]π‘›βˆ’1,𝑏]. By 𝑑 we mean a nonzero derivation in 𝑅.

A well-known result proven by Posner [1] states that if [[𝑑(π‘₯),π‘₯],𝑦]=0 for all π‘₯,π‘¦βˆˆπ‘…, then 𝑅 is commutative. In [2], Lanski generalized this result of Posner to the Lie ideal. Lanski proved that if π‘ˆ is a noncommutative Lie ideal of 𝑅 such that [[𝑑(π‘₯),π‘₯],𝑦]=0 for all π‘₯βˆˆπ‘ˆ,π‘¦βˆˆπ‘…, then either 𝑅 is commutative or char 𝑅=2 and 𝑅 satisfies 𝑆4, the standard identity in four variables. Bell and Martindale III [3] studied this identity for a semiprime ring 𝑅. They proved that if 𝑅 is a semiprime ring and [[𝑑(π‘₯),π‘₯],𝑦]=0 for all π‘₯ in a non-zero left ideal of 𝑅 and π‘¦βˆˆπ‘…, then 𝑅 contains a non-zero central ideal. Clearly, this result says that if 𝑅 is a prime ring, then 𝑅 must be commutative.

Several authors have studied this kind of Engel type identities with derivation in different ways. In [4], Herstein proved that if char 𝑅≠2 and [𝑑(π‘₯),𝑑(𝑦)]=0 for all π‘₯,π‘¦βˆˆπ‘…, then 𝑅 is commutative. In [5], Filippis showed that if 𝑅 is of characteristic different from 2 and 𝜌 a non-zero right ideal of 𝑅 such that [𝜌,𝜌]πœŒβ‰ 0 and [[𝑑(π‘₯),π‘₯],[𝑑(𝑦),𝑦]]=0 for all π‘₯,π‘¦βˆˆπœŒ, then 𝑑(𝜌)𝜌=0.

In continuation of these previous results, it is natural to consider the situation when [[𝑑(π‘₯),π‘₯]𝑛,[𝑦,𝑑(𝑦)]π‘š]𝑑=0 for all π‘₯,π‘¦βˆˆπœŒ, 𝑛,π‘šβ‰₯0,𝑑β‰₯1 are fixed integers. We have studied this identity in the present paper.

It is well known that any derivation of a prime ring 𝑅 can be uniquely extended to a derivation of 𝑄, and so any derivation of 𝑅 can be defined on the whole of 𝑄. Moreover 𝑄 is a prime ring as well as 𝑅 and the extended centroid 𝐢 of 𝑅 coincides with the center of 𝑄. We refer to [6, 7] for more details.

Denote by π‘„βˆ—πΆπΆ{𝑋,π‘Œ} the free product of the 𝐢-algebra 𝑄 and 𝐢{𝑋,π‘Œ}, the free 𝐢-algebra in noncommuting indeterminates 𝑋,π‘Œ.

2. The Case: 𝑅 Prime Ring

We need the following lemma.

Lemma 2.1. Let 𝜌 be a non-zero right ideal of 𝑅 and 𝑑 a derivation of 𝑅. Then the following conditions are equivalent: (i) d is an inner derivation induced by some π‘βˆˆπ‘„ such that π‘πœŒ=0; (ii) 𝑑(𝜌)𝜌=0 (for its proof refer to [8, Lemma]).

We mention an important result which will be used quite frequently as follows.

Theorem 2.2 (see Kharchenko [9]). Let 𝑅 be a prime ring, 𝑑 a derivation on 𝑅 and 𝐼 a non-zero ideal of 𝑅. If 𝐼 satisfies the differential identity 𝑓(π‘Ÿ1,π‘Ÿ2,…,π‘Ÿπ‘›,𝑑(π‘Ÿ1),𝑑(π‘Ÿ2),…,𝑑(π‘Ÿπ‘›))=0foranyπ‘Ÿ1,π‘Ÿ2,…,π‘Ÿπ‘›βˆˆπΌ, then either (i) 𝐼 satisfies the generalized polynomial identity π‘“ξ€·π‘Ÿ1,π‘Ÿ2,…,π‘Ÿπ‘›,π‘₯1,π‘₯2,…,π‘₯𝑛=0,(2.1) or (ii) 𝑑 is 𝑄-inner, that is, for some π‘žβˆˆπ‘„,𝑑(π‘₯)=[π‘ž,π‘₯] and 𝐼 satisfies the generalized polynomial identity π‘“ξ€·π‘Ÿ1,π‘Ÿ2,…,π‘Ÿπ‘›,ξ€Ίπ‘ž,π‘Ÿ1ξ€»,ξ€Ίπ‘ž,π‘Ÿ2ξ€»ξ€Ί,…,π‘ž,π‘Ÿπ‘›ξ€»ξ€Έ=0.(2.2)

Theorem 2.3. Let 𝑅 be a prime ring of char 𝑅≠2 and 𝑑 a derivation of 𝑅 such that [[𝑑(π‘₯),π‘₯]𝑛,[[𝑦,𝑑(𝑦)]π‘š]𝑑=0 for all π‘₯,π‘¦βˆˆπ‘…, where 𝑛β‰₯0,π‘šβ‰₯0,𝑑β‰₯1 are fixed integers. Then 𝑅 is commutative or 𝑑=0.

Proof. Let 𝑅 be noncommutative. If 𝑑 is not 𝑄-inner, then by Kharchenko's Theorem [9] 𝑔[](π‘₯,𝑦,𝑒,𝑣)=𝑒,π‘₯𝑛,[]𝑦,π‘£π‘šξ€»π‘‘=0,(2.3) for all π‘₯,𝑦,𝑒,π‘£βˆˆπ‘…. This is a polynomial identity and hence there exists a field 𝐹 such that π‘…βŠ†π‘€π‘˜(𝐹) with π‘˜>1, and 𝑅 and π‘€π‘˜(𝐹) satisfy the same polynomial identity [10,Lemma 1]. But by choosing 𝑒=𝑒12,π‘₯=𝑒11,𝑣=𝑒11 and 𝑦=𝑒21, we get ξ€Ί[]0=𝑒,π‘₯𝑛,[]𝑦,π‘£π‘šξ€»π‘‘=(βˆ’1)𝑑𝑛𝑒11+(βˆ’)𝑑𝑒22ξ€Έ,(2.4) which is a contradiction.
Now, let 𝑑 be 𝑄-inner derivation, say 𝑑=π‘Žπ‘‘(π‘Ž) for some π‘Žβˆˆπ‘„, that is, 𝑑(π‘₯)=[π‘Ž,π‘₯] for all π‘₯βˆˆπ‘…, then we have ξ€Ί[]π‘Ž,π‘₯𝑛+1,[[𝑦,π‘Ž,𝑦]]π‘šξ€»π‘‘=0,(2.5) for all π‘₯,π‘¦βˆˆπ‘…. Since 𝑑≠0, π‘Žβˆ‰πΆ and hence 𝑅 satisfies a nontrivial generalized polynomial identity (GPI). By [11], it follows that 𝑅𝐢 is a primitive ring with 𝐻=π‘†π‘œπ‘(𝑅𝐢)β‰ 0, and 𝑒𝐻𝑒 is finite dimensional over 𝐢 for any minimal idempotent π‘’βˆˆπ‘…πΆ. Moreover we may assume that 𝐻 is noncommutative; otherwise, 𝑅 must be commutative which is a contradiction.
Notice that 𝐻 satisfies [[π‘Ž,π‘₯]𝑛+1,[𝑦,[π‘Ž,𝑦]]π‘š]𝑑=0 (see [10, Proof of Theorem 1]). For any idempotent π‘’βˆˆπ» and π‘₯∈𝐻, we have ξ€Ί[]0=π‘Ž,𝑒𝑛+1,[[𝑒π‘₯(1βˆ’π‘’),π‘Ž,𝑒π‘₯(1βˆ’π‘’)]]π‘šξ€»π‘‘.(2.6) Right multiplying by 𝑒, we get ξ€Ί[]0=π‘Ž,𝑒𝑛+1,[[𝑒π‘₯(1βˆ’π‘’),π‘Ž,𝑒π‘₯(1βˆ’π‘’)]]π‘šξ€»π‘‘π‘’=ξ€Ί[]π‘Ž,𝑒𝑛+1,[[𝑒π‘₯(1βˆ’π‘’),π‘Ž,𝑒π‘₯(1βˆ’π‘’)]]π‘šξ€»π‘‘βˆ’1β‹…ξ€½[]π‘Ž,𝑒𝑛+1ξ€·[[𝑒π‘₯(1βˆ’π‘’),π‘Ž,𝑒π‘₯(1βˆ’π‘’)]]π‘šξ€Έξ€·[[π‘’βˆ’π‘’π‘₯(1βˆ’π‘’),π‘Ž,𝑒π‘₯(1βˆ’π‘’)]]π‘šξ€Έ[]π‘Ž,𝑒𝑛+1𝑒=ξ€Ί[]π‘Ž,𝑒𝑛+1,[[𝑒π‘₯(1βˆ’π‘’),π‘Ž,𝑒π‘₯(1βˆ’π‘’)]]π‘šξ€»π‘‘βˆ’1β‹…ξƒ―[]π‘Ž,𝑒𝑛+1ξƒ©π‘šξ“π‘—=0(βˆ’1)π‘—ξ‚΅π‘šπ‘—ξ‚Ά[]π‘Ž,𝑒π‘₯(1βˆ’π‘’)𝑗[]𝑒π‘₯(1βˆ’π‘’)π‘Ž,𝑒π‘₯(1βˆ’π‘’)π‘šβˆ’π‘—ξƒͺπ‘’βˆ’ξƒ©π‘šξ“π‘—=0(βˆ’1)π‘—ξ‚΅π‘šπ‘—ξ‚Ά[]π‘Ž,𝑒π‘₯(1βˆ’π‘’)𝑗[]𝑒π‘₯(1βˆ’π‘’)π‘Ž,𝑒π‘₯(1βˆ’π‘’)π‘šβˆ’π‘—ξƒͺ[]π‘Ž,𝑒𝑛+1𝑒=ξ€Ί[]π‘Ž,𝑒𝑛+1,[[𝑒π‘₯(1βˆ’π‘’),π‘Ž,𝑒π‘₯(1βˆ’π‘’)]]π‘šξ€»π‘‘βˆ’1⋅0βˆ’π‘šξ“π‘—=0(βˆ’1)π‘—ξ‚΅π‘šπ‘—ξ‚Ά(βˆ’π‘’π‘₯(1βˆ’π‘’)π‘Ž)𝑗𝑒π‘₯(1βˆ’π‘’)(π‘Žπ‘’π‘₯(1βˆ’π‘’))π‘šβˆ’π‘—ξƒͺξƒ°ξ€Ί[]π‘Žπ‘’=βˆ’π‘Ž,𝑒𝑛+1,[[𝑒π‘₯(1βˆ’π‘’),π‘Ž,𝑒π‘₯(1βˆ’π‘’)]]π‘šξ€»π‘‘βˆ’1ξƒ©π‘šξ“π‘—=0ξ‚΅π‘šπ‘—ξ‚Ά(𝑒π‘₯(1βˆ’π‘’)π‘Ž)π‘š+1ξƒͺ𝑒=βˆ’2π‘šξ€Ί[]π‘Ž,𝑒𝑛+1,[[𝑒π‘₯(1βˆ’π‘’),π‘Ž,𝑒π‘₯(1βˆ’π‘’)]]π‘šξ€»π‘‘βˆ’1(𝑒π‘₯(1βˆ’π‘’)π‘Ž)π‘š+1𝑒=(βˆ’)𝑑2π‘šπ‘‘(𝑒π‘₯(1βˆ’π‘’)π‘Ž)(π‘š+1)𝑑𝑒.(2.7)
This implies that 0=(βˆ’)𝑑2π‘šπ‘‘((1βˆ’π‘’)π‘Žπ‘’π‘₯)(π‘š+1)𝑑+1. Since char 𝑅≠2, ((1βˆ’π‘’)π‘Žπ‘’π‘₯)(π‘š+1)𝑑+1=0. By Levitzki's lemma [12, Lemma 1.1], (1βˆ’π‘’)π‘Žπ‘’π‘₯=0 for all π‘₯∈𝐻. Since 𝐻 is prime ring, (1βˆ’π‘’)π‘Žπ‘’=0, that is, π‘’π‘Žπ‘’=π‘Žπ‘’ for any idempotent π‘’βˆˆπ». Now replacing 𝑒 with 1βˆ’π‘’, we get that π‘’π‘Ž(1βˆ’π‘’)=0, that is, π‘’π‘Žπ‘’=π‘’π‘Ž. Therefore for any idempotent π‘’βˆˆπ», we have [π‘Ž,𝑒]=0. So π‘Ž commutes with all idempotents in 𝐻. Since 𝐻 is a simple ring, either 𝐻 is generated by its idempotents or 𝐻 does not contain any nontrivial idempotents. The first case gives π‘ŽβˆˆπΆ contradicting 𝑑≠0. In the last case, 𝐻 is a finite dimensional division algebra over 𝐢. This implies that 𝐻=𝑅𝐢=𝑄 and π‘Žβˆˆπ». By [10,Lemma 2], there exists a field 𝐹 such that π»βŠ†π‘€π‘˜(𝐹) and π‘€π‘˜(𝐹) satisfies [[π‘Ž,π‘₯]𝑛+1,[𝑦,[π‘Ž,𝑦]]π‘š]𝑑. Then by the same argument as earlier, π‘Ž commutes with all idempotents in π‘€π‘˜(𝐹), again giving the contradiction π‘ŽβˆˆπΆ, that is, 𝑑=0. This completes the proof of the theorem.

Theorem 2.4. Let 𝑅 be a prime ring of char 𝑅≠2, 𝑑 a non-zero derivation of 𝑅 and 𝜌 a non-zero right ideal of 𝑅 such that [[𝑑(π‘₯),π‘₯]𝑛,[𝑦,𝑑(𝑦)]π‘š]𝑑=0 for all π‘₯,π‘¦βˆˆπœŒ, where 𝑛β‰₯0,π‘šβ‰₯0,𝑑β‰₯1 are fixed integers. If [𝜌,𝜌]πœŒβ‰ 0, then 𝑑(𝜌)𝜌=0.

We begin the proof by proving the following lemma.

Lemma 2.5. If 𝑑(𝜌)πœŒβ‰ 0 and [[𝑑(π‘₯),π‘₯]𝑛,[𝑦,𝑑(𝑦)]π‘š]𝑑=0 for all π‘₯,π‘¦βˆˆπœŒ,π‘š,𝑛β‰₯0,𝑑β‰₯1 are fixed integers, then 𝑅 satisfies nontrivial generalized polynomial identity (GPI).

Proof. Suppose on the contrary that 𝑅 does not satisfy any nontrivial GPI. We may assume that 𝑅 is noncommutative; otherwise, 𝑅 satisfies trivially a nontrivial GPI. We consider two cases.Case 1. Suppose that 𝑑 is 𝑄-inner derivation induced by an element π‘Žβˆˆπ‘„. Then for any π‘₯∈𝜌,ξ€Ί[]π‘Ž,π‘₯𝑋𝑛+1,[[π‘₯π‘Œ,π‘Ž,π‘₯π‘Œ]]π‘šξ€»π‘‘(2.8) is a GPI for 𝑅, so it is the zero element in π‘„βˆ—πΆπΆ{𝑋,π‘Œ}. Expanding this, we get []π‘Ž,π‘₯π‘‹π‘šπ‘›+1𝑗=0(βˆ’1)π‘—ξ‚΅π‘šπ‘—ξ‚Ά[]π‘Ž,π‘₯π‘Œπ‘—[]π‘₯π‘Œπ‘Ž,π‘₯π‘Œπ‘šβˆ’π‘—βˆ’π‘šξ“π‘—=0(βˆ’1)π‘—ξ‚΅π‘šπ‘—ξ‚Ά[]π‘Ž,π‘₯π‘Œπ‘—[]π‘₯π‘Œπ‘Ž,π‘₯π‘Œπ‘šβˆ’π‘—[]π‘Ž,π‘₯𝑋𝑛+1ξƒͺ𝐴(𝑋,π‘Œ)=0,(2.9) where 𝐴(𝑋,π‘Œ)=[[π‘Ž,π‘₯𝑋]𝑛+1,[π‘₯π‘Œ,[π‘Ž,π‘₯π‘Œ]]π‘š]π‘‘βˆ’1. If π‘Žπ‘₯ and π‘₯ are linearly 𝐢-independent for some π‘₯∈𝜌, then (π‘Žπ‘₯𝑋)π‘šπ‘›+1𝑗=0(βˆ’1)π‘—ξ‚΅π‘šπ‘—ξ‚Ά[]π‘Ž,π‘₯π‘Œπ‘—[]π‘₯π‘Œπ‘Ž,π‘₯π‘Œπ‘šβˆ’π‘—βˆ’π‘šξ“π‘—=0(βˆ’1)π‘—ξ‚΅π‘šπ‘—ξ‚Ά(π‘Žπ‘₯π‘Œ)𝑗[]π‘₯π‘Œπ‘Ž,π‘₯π‘Œπ‘šβˆ’π‘—[]π‘Ž,π‘₯𝑋𝑛+1ξƒͺ𝐴(𝑋,π‘Œ)=0.(2.10) Again, since π‘Žπ‘₯ and π‘₯ are linearly 𝐢-independent, above relation implies that ξ€·[]βˆ’π‘₯π‘Œπ‘Ž,π‘₯π‘Œπ‘š[]π‘Ž,π‘₯𝑋𝑛+1𝐴(𝑋,π‘Œ)=0,(2.11) and so ξ€·βˆ’π‘₯π‘Œ(π‘Žπ‘₯π‘Œ)π‘š(π‘Žπ‘₯𝑋)𝑛+1𝐴(𝑋,π‘Œ)=0.(2.12) Repeating the same process yields ξ€·βˆ’π‘₯π‘Œ(π‘Žπ‘₯π‘Œ)π‘š(π‘Žπ‘₯𝑋)𝑛+1𝑑=0(2.13) in π‘„βˆ—πΆπΆ{𝑋,π‘Œ}. This implies that π‘Žπ‘₯=0, a contradiction. Thus for any π‘₯∈𝜌, π‘Žπ‘₯ and π‘₯ are 𝐢-dependent. Then (π‘Žβˆ’π›Ό)𝜌=0 for some π›ΌβˆˆπΆ. Replacing π‘Ž with π‘Žβˆ’π›Ό, we may assume that π‘ŽπœŒ=0. Then by Lemma 2.1, 𝑑(𝜌)𝜌=0, contradiction.
Case 2. Suppose that 𝑑 is not 𝑄-inner derivation. If for all π‘₯∈𝜌, 𝑑(π‘₯)∈π‘₯𝐢, then [𝑑(π‘₯),π‘₯]=0 which implies that 𝑅 is commutative (see [13]). Therefore there exists π‘₯∈𝜌 such that 𝑑(π‘₯)βˆ‰π‘₯𝐢, that is, π‘₯ and 𝑑(π‘₯) are linearly 𝐢-independent.
By our assumption, we have that 𝑅 satisfies
ξ€Ί[𝑑](π‘₯𝑋),π‘₯𝑋𝑛,[]π‘₯π‘Œ,𝑑(π‘₯π‘Œ)π‘šξ€»π‘‘=0.(2.14) By Kharchenko's Theorem [9], 𝑑(π‘₯)𝑋+π‘₯π‘Ÿ1ξ€»,π‘₯𝑋𝑛,ξ€Ίπ‘₯π‘Œ,𝑑(π‘₯)π‘Œ+π‘₯π‘Ÿ2ξ€»π‘šξ€»π‘‘=0,(2.15) for all 𝑋,π‘Œ,π‘Ÿ1,π‘Ÿ2βˆˆπ‘…. In particular for π‘Ÿ1=π‘Ÿ2=0, ξ€Ί[𝑑](π‘₯)𝑋,π‘₯𝑋𝑛,[]π‘₯π‘Œ,𝑑(π‘₯)π‘Œπ‘šξ€»π‘‘=0,(2.16) which is a nontrivial GPI for 𝑅, because π‘₯ and 𝑑(π‘₯) are linearly 𝐢-independent, a contradiction.

We are now ready to prove our main theorem.

Proof of Theorem 2.4. Suppose that 𝑑(𝜌)πœŒβ‰ 0, then we derive a contradiction. By Lemma 2.5, 𝑅 is a prime GPI ring, so is also 𝑄 by [14]. Since 𝑄 is centrally closed over 𝐢, it follows from [11] that 𝑄 is a primitive ring with 𝐻=π‘†π‘œπ‘(𝑄)β‰ 0.
By our assumption and by [7], we may assume that ξ€Ί[𝑑](π‘₯),π‘₯𝑛,[]𝑦,𝑑(𝑦)π‘šξ€»π‘‘=0(2.17) is satisfied by πœŒπ‘„ and hence by 𝜌𝐻. Let 𝑒=𝑒2∈𝜌𝐻 and π‘¦βˆˆπ». Then replacing π‘₯ with 𝑒 and 𝑦 with 𝑒𝑦(1βˆ’π‘’) in (2.17), then right multiplying it by 𝑒, we obtain that ξ€Ί[𝑑]0=(𝑒),𝑒𝑛,[]𝑒𝑦(1βˆ’π‘’),𝑑(𝑒𝑦(1βˆ’π‘’))π‘šξ€»π‘‘π‘’=ξ€Ί[𝑑](𝑒),𝑒𝑛,[]𝑒𝑦(1βˆ’π‘’),𝑑(𝑒𝑦(1βˆ’π‘’))π‘šξ€»π‘‘βˆ’1β‹…ξƒ―[]𝑑(𝑒),π‘’π‘›π‘šξ“π‘—=0(βˆ’1)π‘—ξ‚΅π‘šπ‘—ξ‚Άπ‘‘(𝑒𝑦(1βˆ’π‘’))𝑗𝑒𝑦(1βˆ’π‘’)𝑑(𝑒𝑦(1βˆ’π‘’))π‘šβˆ’π‘—π‘’βˆ’π‘šξ“π‘—=0(βˆ’1)π‘—ξ‚΅π‘šπ‘—ξ‚Άπ‘‘(𝑒𝑦(1βˆ’π‘’))𝑗𝑒𝑦(1βˆ’π‘’)𝑑(𝑒𝑦(1βˆ’π‘’))π‘šβˆ’π‘—[]𝑑(𝑒),𝑒𝑛𝑒.(2.18)
Now we have the fact that for any idempotent 𝑒, 𝑑(𝑦(1βˆ’π‘’))𝑒=βˆ’π‘¦(1βˆ’π‘’)𝑑(𝑒), 𝑒𝑑(𝑒)𝑒=0 and so ξ€Ί[]0=𝑑(𝑒),𝑒𝑛,[]𝑒𝑦(1βˆ’π‘’),𝑑(𝑒𝑦(1βˆ’π‘’))π‘šξ€»π‘‘βˆ’1β‹…ξƒ―0βˆ’π‘šξ“π‘—=0(βˆ’1)π‘—ξ‚΅π‘šπ‘—ξ‚Άπ‘’(βˆ’π‘¦(1βˆ’π‘’)𝑑(𝑒))𝑗𝑦(1βˆ’π‘’)𝑑(𝑒𝑦(1βˆ’π‘’))π‘šβˆ’π‘—ξƒ°.𝑑(𝑒)𝑒(2.19) Now since for any idempotent 𝑒 and for any π‘¦βˆˆπ‘…, (1βˆ’π‘’)𝑑(𝑒𝑦)=(1βˆ’π‘’)𝑑(𝑒)𝑦, above relation gives ξ€Ί[]0=𝑑(𝑒),𝑒𝑛,[]𝑒𝑦(1βˆ’π‘’),𝑑(𝑒𝑦(1βˆ’π‘’))π‘šξ€»π‘‘βˆ’1β‹…ξƒ―βˆ’π‘’π‘šξ“π‘—=0ξ‚΅π‘šπ‘—ξ‚Ά(𝑦(1βˆ’π‘’)𝑑(𝑒))𝑗𝑦(1βˆ’π‘’)(𝑑(𝑒)𝑦(1βˆ’π‘’))π‘šβˆ’π‘—ξƒ°=ξ€Ί[]𝑑(𝑒)𝑒𝑑(𝑒),𝑒𝑛,[]𝑒𝑦(1βˆ’π‘’),𝑑(𝑒𝑦(1βˆ’π‘’))π‘šξ€»π‘‘βˆ’1ξƒ―βˆ’π‘’π‘šξ“π‘—=0ξ‚΅π‘šπ‘—ξ‚Ά(𝑦(1βˆ’π‘’)𝑑(𝑒))π‘š+1𝑒=ξ€Ί[]𝑑(𝑒),𝑒𝑛,[]𝑒𝑦(1βˆ’π‘’),𝑑(𝑒𝑦(1βˆ’π‘’))π‘šξ€»π‘‘βˆ’1ξ€½βˆ’2π‘šπ‘’(𝑦(1βˆ’π‘’)𝑑(𝑒))π‘š+1𝑒=ξ€½βˆ’2π‘šπ‘’(𝑦(1βˆ’π‘’)𝑑(𝑒))π‘š+1𝑑𝑒.(2.20) This implies that 0=(βˆ’1)𝑑2π‘šπ‘‘((1βˆ’π‘’)𝑑(𝑒)𝑒𝑦)(π‘š+1)𝑑+1 for all π‘¦βˆˆπ». Since char 𝑅≠2, we have by Levitzki's lemma [12,Lemma 1.1] that (1βˆ’π‘’)𝑑(𝑒)𝑒𝑦=0 for all π‘¦βˆˆπ». By primeness of 𝐻, (1βˆ’π‘’)𝑑(𝑒)𝑒=0. By [15,Lemma 1], since 𝐻 is a regular ring, for each π‘ŸβˆˆπœŒπ», there exists an idempotent π‘’βˆˆπœŒπ» such that π‘Ÿ=π‘’π‘Ÿ and π‘’βˆˆπ‘Ÿπ». Hence (1βˆ’π‘’)𝑑(𝑒)𝑒=0 gives (1βˆ’π‘’)𝑑(𝑒)=(1βˆ’π‘’)𝑑(𝑒2)=(1βˆ’π‘’)𝑑(𝑒)𝑒=0 and so 𝑑(𝑒)=𝑒𝑑(𝑒)βˆˆπ‘’π»βŠ†πœŒπ» and 𝑑(π‘Ÿ)=𝑑(π‘’π‘Ÿ)=𝑑(𝑒)π‘’π‘Ÿ+𝑒𝑑(π‘’π‘Ÿ)∈𝜌𝐻. Hence for each π‘ŸβˆˆπœŒπ», 𝑑(π‘Ÿ)∈𝜌𝐻. Thus 𝑑(𝜌𝐻)βŠ†πœŒπ». Set 𝐽=𝜌𝐻. Then 𝐽=𝐽/(π½βˆ©π‘™π»(𝐽)), a prime 𝐢-algebra with the derivation 𝑑 such that 𝑑(π‘₯)=𝑑(π‘₯), for all π‘₯∈𝐽. By assumption, we have that 𝑑π‘₯ξ€Έ,π‘₯𝑛,𝑦,π‘‘ξ€·π‘¦ξ€Έξ‚„π‘šξ‚„π‘‘=0,(2.21) for all π‘₯,π‘¦βˆˆπ½. By Theorem 2.3, we have either 𝑑=0 or 𝜌𝐻 is commutative. Therefore we have that either 𝑑(𝜌𝐻)𝜌𝐻=0 or [𝜌𝐻,𝜌𝐻]𝜌𝐻=0. Now 𝑑(𝜌𝐻)𝜌𝐻=0 implies that 0=𝑑(𝜌𝜌𝐻)𝜌𝐻=𝑑(𝜌)𝜌𝐻𝜌𝐻 and so 𝑑(𝜌)𝜌=0. [𝜌𝐻,𝜌𝐻]𝜌𝐻=0 implies that 0=[𝜌𝜌𝐻,𝜌𝐻]𝜌𝐻=[𝜌,𝜌𝐻]𝜌𝐻𝜌𝐻 and so [𝜌,𝜌𝐻]𝜌=0, then 0=[𝜌,𝜌𝜌𝐻]𝜌=[𝜌,𝜌]𝜌𝐻𝜌 implying that [𝜌,𝜌]𝜌=0. Thus in all the cases we have contradiction. This completes the proof of the theorem.

3. The Case: 𝑅 Semiprime Ring

In this section we extend Theorem 2.3 to the semiprime case. Let 𝑅 be a semiprime ring and π‘ˆ be its right Utumi quotient ring. It is well known that any derivation of a semiprime ring 𝑅 can be uniquely extended to a derivation of its right Utumi quotient ring π‘ˆ and so any derivation of 𝑅 can be defined on the whole of π‘ˆ [7,Lemma 2].

By the standard theory of orthogonal completions for semiprime rings, we have the following lemma.

Lemma 3.1 (see [16, Lemma 1 and Theorem 1] or [7,pages 31-32]). Let 𝑅 be a 2-torsion free semiprime ring and 𝑃 a maximal ideal of 𝐢. Then π‘ƒπ‘ˆ is a prime ideal of π‘ˆ invariant under all derivations of π‘ˆ. Moreover, β‹‚{π‘ƒπ‘ˆβˆ£π‘ƒisamaximalidealof𝐢withπ‘ˆ/π‘ƒπ‘ˆ2-torsionfree}=0.

Theorem 3.2. Let 𝑅 be a 2-torsion free semiprime ring and 𝑑 a non-zero derivation of 𝑅 such that [[𝑑(π‘₯),π‘₯]𝑛,[𝑦,𝑑(𝑦)]π‘š]𝑑=0 for all π‘₯,π‘¦βˆˆπ‘…, 𝑛,π‘šβ‰₯0,𝑑β‰₯1 fixed are integers. Then 𝑑 maps 𝑅 into its center.

Proof. Since any derivation 𝑑 can be uniquely extended to a derivation in π‘ˆ, and 𝑅 and π‘ˆ satisfy the same differential identities [7, Theorem 3], we have ξ€Ί[𝑑](π‘₯),π‘₯𝑛,[]𝑦,𝑑(𝑦)π‘šξ€»π‘‘=0,(3.1) for all π‘₯,π‘¦βˆˆπ‘ˆ. Let 𝑃 be any maximal ideal of 𝐢 such that π‘ˆ/π‘ƒπ‘ˆ is 2-torsion free. Then by Lemma 3.1, π‘ƒπ‘ˆ is a prime ideal of π‘ˆ invariant under 𝑑. Set π‘ˆ=π‘ˆ/π‘ƒπ‘ˆ. Then derivation 𝑑 canonically induces a derivation 𝑑 on π‘ˆ defined by 𝑑(π‘₯)=𝑑(π‘₯) for all π‘₯βˆˆπ‘ˆ. Therefore, 𝑑π‘₯ξ€Έ,π‘₯𝑛,𝑦,π‘‘ξ€·π‘¦ξ€Έξ‚„π‘šξ‚„π‘‘=0,(3.2) for all π‘₯,π‘¦βˆˆπ‘ˆ. By Theorem 2.3, either 𝑑=0 or [π‘ˆ,π‘ˆ]=0, that is, 𝑑(π‘ˆ)βŠ†π‘ƒπ‘ˆ or [π‘ˆ,π‘ˆ]βŠ†π‘ƒπ‘ˆ. In any case 𝑑(π‘ˆ)[π‘ˆ,π‘ˆ]βŠ†π‘ƒπ‘ˆ for any maximal ideal 𝑃 of 𝐢. By Lemma 3.1, β‹‚{π‘ƒπ‘ˆβˆ£π‘ƒisamaximalidealof𝐢withπ‘ˆ/π‘ƒπ‘ˆ2-torsionfree}=0. Thus 𝑑(π‘ˆ)[π‘ˆ,π‘ˆ]=0. Without loss of generality, we have 𝑑(𝑅)[𝑅,𝑅]=0. This implies that 𝑅0=𝑑2ξ€Έ[][][][].𝑅,𝑅=𝑑(𝑅)𝑅𝑅,𝑅+𝑅𝑑(𝑅)𝑅,𝑅=𝑑(𝑅)𝑅𝑅,𝑅(3.3) Therefore [𝑅,𝑑(𝑅)]𝑅[𝑅,𝑑(𝑅)]=0. By semiprimeness of 𝑅, we have [𝑅,𝑑(𝑅)]=0, that is, 𝑑(𝑅)βŠ†π‘(𝑅). This completes the proof of the theorem.

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