International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

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Volume 2011 |Article ID 750382 | https://doi.org/10.5402/2011/750382

Basudeb Dhara, Atanu Pattanayak, "Generalized Derivations and Left Ideals in Prime and Semiprime Rings", International Scholarly Research Notices, vol. 2011, Article ID 750382, 5 pages, 2011. https://doi.org/10.5402/2011/750382

Generalized Derivations and Left Ideals in Prime and Semiprime Rings

Academic Editor: A. Rapinchuk
Received19 May 2011
Accepted07 Jul 2011
Published15 Aug 2011

Abstract

Let 𝑅 be an associative ring, 𝜆 a nonzero left ideal of 𝑅, 𝑑∶𝑅→𝑅 a derivation and 𝐺∶𝑅→𝑅 a generalized derivation. In this paper, we study the following situations in prime and semiprime rings: (1) 𝐺(𝑥∘𝑦)=ğ‘Ž(xy±yx); (2) 𝐺[𝑥,𝑦]=ğ‘Ž(xy±yx); (3) 𝑑(𝑥)∘𝑑(𝑦)=ğ‘Ž(xy±yx); for all 𝑥,𝑦∈𝜆 and ğ‘Žâˆˆ{0,1,−1}.

1. Introduction

Throughout this paper, let 𝑅 be an associative ring, 𝜆 a left ideal of 𝑅, 𝑑 a derivation of 𝑅 and 𝐺 a generalized derivation of 𝑅. For any two elements 𝑥,𝑦∈𝑅, [𝑥,𝑦] will denote the commutator element 𝑥𝑦−𝑦𝑥 and 𝑥∘𝑦 denotes 𝑥𝑦+𝑦𝑥. We use extensively the following basic commutator identities: [𝑥𝑦,𝑧]=[𝑥,𝑧]𝑦+𝑥[𝑦,𝑧] and [𝑥,𝑦𝑧]=[𝑥,𝑦]𝑧+𝑦[𝑥,𝑧]. Recall that a ring 𝑅 is called prime, if for any ğ‘Ž,𝑏∈𝑅, ğ‘Žğ‘…ğ‘=(0) implies that either ğ‘Ž=0 or 𝑏=0 and is called semiprime if for any ğ‘Žâˆˆğ‘…, ğ‘Žğ‘…ğ‘Ž=(0) implies ğ‘Ž=0. An additive mapping 𝑑∶𝑅→𝑅 is said to be a derivation of 𝑅 if for any 𝑥,𝑦∈𝑅, 𝑑(𝑥𝑦)=𝑑(𝑥)𝑦+𝑥𝑑(𝑦) holds. The generalized derivation of 𝑅 is defined as an additive mapping 𝐺∶𝑅→𝑅 such that 𝐺(𝑥𝑦)=𝐺(𝑥)𝑦+𝑥𝑑(𝑦) holds for any 𝑥,𝑦∈𝑅, where 𝑑 is a derivation of 𝑅. So, every derivation is a generalized derivation, but the converse is not true in general. If 𝑑=0, then we have 𝐺(𝑥𝑦)=𝐺(𝑥)𝑦 for all 𝑥,𝑦∈𝑅, which is called a left multiplier mapping of 𝑅. Thus, generalized derivation generalizes both the concepts, derivation as well as left multiplier mapping of 𝑅.

In [1], Daif and Bell proved that if 𝑅 is a semiprime ring with a nonzero ideal 𝐼 and 𝑑 is a derivation of 𝑅 such that 𝑑([𝑥,𝑦])=±[𝑥,𝑦] for all 𝑥,𝑦∈𝐼, then 𝐼 is central ideal. In particular, if 𝐼=𝑅, then 𝑅 is commutative. Recently, Quadri et al. [2] have generalized this result replacing derivation 𝑑 with a generalized derivation in a prime ring 𝑅. More precisely, they obtained the following result.

Let 𝑅 be a prime ring and 𝐼 a nonzero ideal of 𝑅. If 𝑅 admits a generalized derivation 𝐹 associated with a nonzero derivation 𝑑 such that any one of the following holds: (i) 𝐹([𝑥,𝑦])=[𝑥,𝑦] for all 𝑥,𝑦∈𝐼; (ii) 𝐹([𝑥,𝑦])=−[𝑥,𝑦] for all 𝑥,𝑦∈𝐼; (iii) 𝐹(𝑥∘𝑦)=(𝑥∘𝑦) for all 𝑥,𝑦∈𝐼; (iv) 𝐹(𝑥∘𝑦)=−(𝑥∘𝑦) for all 𝑥,𝑦∈𝐼; then 𝑅 is commutative.

Recently in [3], the first author of this paper has studied all the results of [2] in semiprime ring. In the present paper, our aim is to discuss similar identities in a left sided ideal of a semiprime rings.

2. Main Results

Theorem 2.1. Let 𝑅 be a semiprime ring and 𝜆 a nonzero left ideal of 𝑅. If G is a generalized derivation of 𝑅 associated with a derivation 𝑑 of 𝑅 such that 𝐺(𝑥∘𝑦)=ğ‘Ž(𝑥±𝑦) for all 𝑥,𝑦∈𝜆, where ğ‘Žâˆˆ{0,1,−1}, then [𝜆,𝜆]𝑑(𝜆)=0.

Proof. If 𝐺(𝜆)=0, then for any 𝑥,𝑦∈𝜆, 0=𝐺(𝑥𝑦)=𝐺(𝑥)𝑦+𝑥𝑑(𝑦)=𝑥𝑑(𝑦),(2.1) that is, 𝜆𝑑(𝜆)=0. This gives our conclusion. So let 𝐺(𝜆)≠0. Then by our assumption, we have 𝐺(𝑥∘𝑦)=ğ‘Ž(𝑥±𝑦)(2.2) for all 𝑥,𝑦∈𝜆. Putting 𝑦=𝑦𝑥, 𝑥∈𝜆, we obtain that 𝐺((𝑥∘𝑦)𝑥)=ğ‘Ž((𝑥±𝑦)𝑥). Since 𝐺 is a generalized derivation of 𝑅, this implies that 𝐺(𝑥∘𝑦)𝑥+(𝑥∘𝑦)𝑑(𝑥)=ğ‘Ž(𝑥±𝑦)𝑥. This gives by using (2.2) that (𝑥∘𝑦)𝑑(𝑥)=0 for all 𝑥,𝑦,∈𝜆. Now, we replace 𝑦 with 𝑧𝑦, where 𝑧∈𝜆, and then we get [][]0=𝑧(𝑥∘𝑦)𝑑(𝑥)+𝑥,𝑧𝑦𝑑(𝑥)=𝑥,𝑧𝑦𝑑(𝑥)(2.3) for all 𝑥,𝑦,𝑧∈𝜆. Since 𝜆 is a left ideal, it follows that [𝑥,𝑧]𝑅𝑦𝑑(𝑥)=0 for all 𝑥,𝑦,𝑧∈𝜆. Since 𝑅 is semiprime, it must contain a family Ω={𝑃𝛼∶𝛼∈Λ} of prime ideals such that ⋂𝛼∈Λ𝑃𝛼={0}. If 𝑃 is typical member of Ω and 𝑥∈𝜆, we have either [𝑥,𝜆]⊆𝑃 or 𝜆𝑑(𝑥)⊆𝑃.
For fixed 𝑃, the sets 𝑇1={𝑥∈𝜆∶[𝑥,𝜆]⊆𝑃} and 𝑇2={𝑥∈𝜆∶𝜆𝑑(𝑥)⊆𝑃} form two additive subgroups of 𝜆 such that 𝑇1⋃𝑇2=𝜆. Therefore, either 𝑇1=𝜆 or 𝑇2=𝜆, that is, either [𝜆,𝜆]⊆𝑃 or 𝜆𝑑(𝜆)⊆𝑃. Both of these two conditions together imply that [𝜆,𝜆]𝑑(𝜆)⊆𝑃 for any 𝑃∈Ω. Therefore, ⋂[𝜆,𝜆]𝑑(𝜆)⊆𝛼∈Λ𝑃𝛼=0.

Corollary 2.2. Let 𝑅 be a prime ring and 𝜆 be a nonzero left ideal of 𝑅. If 𝑅 admits a generalized derivation 𝐺 associated with a derivation 𝑑 such that 𝐺(𝑥∘𝑦)=ğ‘Ž(𝑥∘𝑦) for all 𝑥,𝑦∈𝜆, where ğ‘Žâˆˆ{0,1,−1}, then one of the following holds: (i)𝜆𝑑(𝜆)=0;(ii)𝑅 is commutative ring with char(𝑅)=2;(iii)𝑅 is commutative ring with char(𝑅)≠2 and 𝐺(𝑥)=ğ‘Žğ‘¥ for all 𝑥∈𝜆.

Proof. By Theorem 2.1, we have [𝜆,𝜆]𝑑(𝜆)=0. This gives 0=𝜆,𝜆2𝑑[][](𝜆)=𝜆,𝜆𝜆𝑑(𝜆)=𝜆,𝜆𝑅𝜆𝑑(𝜆).(2.4) Since 𝑅 is prime, either [𝜆,𝜆]=0 or 𝜆𝑑(𝜆)=0. Now 𝜆𝑑(𝜆)=0 gives our conclusion (i). So, let [𝜆,𝜆]=0 which gives 0=[𝜆,𝑅𝜆]=[𝜆,𝑅]𝜆 implying 0=[𝜆,𝑅]. Again, this gives 0=[𝑅𝜆,𝑅]=[𝑅,𝑅]𝜆. Since left annihilator of a left-sided ideal is zero, [𝑅,𝑅]=0, that is, 𝑅 is commutative. If char(𝑅)=2, we obtain conclusion (ii). So assume that char(𝑅)≠2. Then our assumption 𝐺(𝑥∘𝑦)=ğ‘Ž(𝑥∘𝑦) gives 2𝐺(𝑥𝑦)=2ğ‘Ž(𝑥𝑦) for all 𝑥,𝑦∈𝜆. Since char(𝑅)≠2, then 𝐺(𝑥𝑦)=ğ‘Ž(𝑥𝑦) for all 𝑥,𝑦∈𝜆. This gives for all 𝑥,𝑦∈𝜆, 0=𝐺(𝑥𝑦)âˆ’ğ‘Ž(𝑥𝑦)=𝐺(𝑥)𝑦+𝑥𝑑(𝑦)âˆ’ğ‘Žğ‘¥ğ‘¦=(𝐺(𝑥)âˆ’ğ‘Žğ‘¥)𝑦+𝑥𝑑(𝑦).(2.5) Let 𝑟∈𝑅. Since 𝑅 is commutative, 𝑥𝑟∈𝜆. Put 𝑥=𝑥𝑟 in last result, we get 0=(𝐺(𝑥)âˆ’ğ‘Žğ‘¥)𝑟𝑦+𝑥𝑟𝑑(𝑦)+𝑥𝑑(𝑟)𝑦={(𝐺(𝑥)âˆ’ğ‘Žğ‘¥)𝑦+𝑥𝑑(𝑦)}𝑟+𝑑(𝑟)𝑦𝑥. Since 𝑅 is commutative, using (2.5), it yields 𝑑(𝑅)𝜆2={0}, implying 𝑑=0. Then, (2.5) implies (𝐺(𝑥)âˆ’ğ‘Žğ‘¥)𝜆=0, which yields 𝐺(𝑥)âˆ’ğ‘Žğ‘¥=0 for all 𝑥∈𝜆.

Theorem 2.3. Let 𝑅 be a semiprime ring, 𝜆 a nonzero left ideal of 𝑅 and 𝐺 a generalized derivation of 𝑅 associated with a derivation 𝑑 of 𝑅. If 𝐺[𝑥,𝑦]=ğ‘Ž(𝑥𝑦±𝑦𝑥) for all 𝑥,𝑦∈𝜆, then [𝜆,𝜆]𝑑(𝜆)=0.

Proof. If 𝐺(𝜆)=0, then 0=𝐺(𝜆2)=𝐺(𝜆)𝜆+𝜆𝑑(𝜆)=𝜆𝑑(𝜆) and hence, [𝜆,𝜆]𝑑(𝜆)=0, which is our conclusion. Assume next that 𝐺(𝜆)≠0. Then by our assumption, we have 𝐺[]𝑥,𝑦=ğ‘Ž(𝑥𝑦±𝑦𝑥)(2.6) for all 𝑥,𝑦∈𝜆. Put 𝑦=𝑦𝑥 and get 𝐺([𝑥,𝑦]𝑥)=ğ‘Ž(𝑥𝑦±𝑦𝑥)𝑥, that is, 𝐺([𝑥,𝑦])𝑥+[𝑥,𝑦]𝑑(𝑥)=ğ‘Ž(𝑥𝑦±𝑦𝑥)𝑥. Now using (2.6), the above relation yields [𝑥,𝑦]𝑑(𝑥)=0 for all 𝑥,𝑦∈𝜆. Putting 𝑦=𝑧𝑦, where 𝑧∈𝜆, we obtain that [𝑥,𝑧]𝑦𝑑(𝑥)=0, which is same as (2.3) in Theorem 2.1. By same argument of Theorem 2.1, we can conclude the result here.

Corollary 2.4. Let 𝑅 be a prime ring and 𝜆 a nonzero left ideal of 𝑅. If 𝐺 is a generalized derivation of 𝑅 associated with a derivation 𝑑 of 𝑅 such that 𝐺[𝑥,𝑦]=ğ‘Ž(𝑥𝑦±𝑦𝑥) for all 𝑥,𝑦∈𝜆, where ğ‘Ž={0,1,−1}, then either 𝑅 is commutative or 𝜆𝑑(𝜆)=0 and one of the following holds: (i)𝜆[𝜆,𝜆]=0;(ii)𝐺(𝑥)=âˆ“ğ‘Žğ‘¥ for all 𝑥∈𝜆. In case 𝐺(𝑥)=âˆ’ğ‘Žğ‘¥ for all 𝑥∈𝜆, with ğ‘Žâ‰ 0, then char(𝑅)=2.

Proof. By the Theorem 2.3, we may conclude that [𝜆,𝜆]𝑑(𝜆)=0. Then by same argument as given in Corollary 2.2, we obtain that either 𝑅 is commutative or 𝜆𝑑(𝜆)=0. Let 𝑅 be noncommutative, then for any 𝑥,𝑦∈𝜆, we have 𝐺(𝑥𝑦)=𝐺(𝑥)𝑦+𝑥𝑑(𝑦)=𝐺(𝑥)𝑦, that is, 𝐺 acts as a left multiplier map on 𝜆. Then for any 𝑥,𝑦,𝑧∈𝜆, replacing 𝑦 with 𝑦𝑧 in our hypothesis 𝐺[𝑥,𝑦]=ğ‘Ž(𝑥𝑦±𝑦𝑥), we have [][][]}𝐺(𝑥,𝑦𝑧+𝑦𝑥,𝑧)=ğ‘Ž{(𝑥𝑦±𝑦𝑥)𝑧∓𝑦𝑥,𝑧(2.7) for all 𝑥,𝑦∈𝜆. Since 𝐺 acts as a left multiplier map on 𝜆, this implies [][][].𝐺(𝑥,𝑦)𝑧+𝐺(𝑦)𝑥,𝑧=ğ‘Ž(𝑥𝑦±𝑦𝑥)ğ‘§âˆ“ğ‘Žğ‘¦ğ‘¥,𝑧(2.8) By using 𝐺[𝑥,𝑦]=ğ‘Ž(𝑥𝑦±𝑦𝑥), it gives 𝐺(𝑦)[𝑥,𝑧]=âˆ“ğ‘Žğ‘¦[𝑥,𝑧], that is, (𝐺(𝑦)Â±ğ‘Žğ‘¦)[𝑥,𝑧]=0 for all 𝑥,𝑦∈𝜆. Replacing 𝑦 with 𝑦𝑢, where 𝑢∈𝜆, we find that (𝐺(𝑦)Â±ğ‘Žğ‘¦)𝑢[𝑥,𝑧]=0, which gives (𝐺(𝑦)Â±ğ‘Žğ‘¦)𝑅𝜆[𝑥,𝑧]=0. Since 𝑅 is prime, either 𝜆[𝜆,𝜆]=0 or 𝐺(𝑦)=âˆ“ğ‘Žğ‘¦ for all 𝑦∈𝜆. When 𝐺(𝑦)=âˆ’ğ‘Žğ‘¦, our assumption 𝐺[𝑥,𝑦]=ğ‘Ž(𝑥𝑦+𝑦𝑥) implies âˆ’ğ‘Ž[𝑥,𝑦]=ğ‘Ž(𝑥𝑦+𝑦𝑥) for all 𝑥,𝑦∈𝜆. This implies 2ğ‘Žğ‘¥ğ‘¦=0, that is, 2ğ‘Žğ‘…ğœ†2=0 implies char(𝑅)=2, unless ğ‘Ž=0.

Theorem 2.5. Let 𝑅 be a semiprime ring, 𝜆 a nonzero left ideal of 𝑅 and 𝑑 a derivation of 𝑅. If 𝑑(𝑥)∘𝑑(𝑦)=ğ‘Ž(𝑥𝑦±𝑦𝑥) for all 𝑥,𝑦∈𝜆, where ğ‘Žâˆˆ{0,1,−1}, then 𝜆[𝑥,𝑑(𝑥)]2=0. In case 𝜆=𝑅 and 𝑅 is 2-torsion free, 𝑑 maps 𝑅 into its center.

Proof. We have for all 𝑥,𝑦∈𝜆, 𝑑(𝑥)𝑑(𝑦)+𝑑(𝑦)𝑑(𝑥)=ğ‘Ž(𝑥𝑦±𝑦𝑥).(2.9) Putting 𝑦=𝑦𝑥, we get 𝑑(𝑥)(𝑑(𝑦)𝑥+𝑦𝑑(𝑥))+(𝑑(𝑦)𝑥+𝑦𝑑(𝑥))𝑑(𝑥)=ğ‘Ž(𝑥𝑦±𝑦𝑥)𝑥.(2.10) Using (2.9), we have []𝑑(𝑥)𝑦𝑑(𝑥)+𝑑(𝑦)𝑥,𝑑(𝑥)+𝑦𝑑(𝑥)2=0.(2.11) Put 𝑦=𝑥𝑦 in (2.11), and get []𝑑(𝑥)𝑥𝑦𝑑(𝑥)+{𝑥𝑑(𝑦)+𝑑(𝑥)𝑦}𝑥,𝑑(𝑥)+𝑥𝑦𝑑(𝑥)2=0.(2.12) Left multiplying (2.11) by 𝑥 and then subtracting from (2.12) yields [][]𝑑(𝑥),𝑥𝑦𝑑(𝑥)+𝑑(𝑥)𝑦𝑥,𝑑(𝑥)=0(2.13) for all 𝑥,𝑦∈𝜆. Put 𝑦=𝑑(𝑥)𝑦 in (2.13), we have []𝑑(𝑥),𝑥𝑑(𝑥)𝑦𝑑(𝑥)+𝑑(𝑥)2𝑦[]𝑥,𝑑(𝑥)=0.(2.14) Left multiplying (2.13) by 𝑑(𝑥) and then subtracting from (2.14), we get [𝑥,𝑑(𝑥)]2𝑦𝑑(𝑥)=0 for all 𝑥,𝑦∈𝜆. This implies [𝑥,𝑑(𝑥)]2𝑦[𝑥,𝑑(𝑥)]2=0 and hence, 𝑦[𝑥,𝑑(𝑥)]2𝑅𝑦[𝑥,𝑑(𝑥)]2=0 for all 𝑥,𝑦∈𝜆. Since 𝑅 is semiprime, 𝜆[𝑥,𝑑(𝑥)]2=0. In case 𝜆=𝑅, [𝑥,𝑑(𝑥)]2=0 for all 𝑥∈𝑅, and then by [4], 𝑑(𝑅)⊆𝑍(𝑅).

References

  1. M. N. Daif and H. E. Bell, “Remarks on derivations on semiprime rings,” International Journal of Mathematics and Mathematical Sciences, vol. 15, no. 1, pp. 205–206, 1992. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  2. M. A. Quadri, M. S. Khan, and N. Rehman, “Generalized derivations and commutativity of prime rings,” Indian Journal of Pure and Applied Mathematics, vol. 34, no. 9, pp. 1393–1396, 2003. View at: Google Scholar | Zentralblatt MATH
  3. B. Dhara, “Remarks on generalized derivations in prime and semiprime rings,” International Journal of Mathematics and Mathematical Sciences, vol. 2010, Article ID 646587, 6 pages, 2010. View at: Publisher Site | Google Scholar
  4. M. T. Koşan, T.-K. Lee, and Y. Zhou, “Identities with Engel conditions on derivations,” to appear in Monatshefte für Mathematik. View at: Publisher Site | Google Scholar

Copyright © 2011 Basudeb Dhara and Atanu Pattanayak. 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.


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