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Communications in AlgebraPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lagb20

On Lie Ideals and Generalized ( θ Φ Φ )-Derivationsin Prime RingsMohammad Ashraf a c , Asma Ali b & Shakir Ali ba Department of Mathematics, Faculty of Science , King Abdulaziz University , Jeddah,Saudi Arabiab Department of Mathematics , Aligarh Muslim University , Aligarh, Indiac Department of Mathematics , Aligarh Muslim University , Aligarh, 202002, IndiaPublished online: 31 Aug 2006.

To cite this article: Mohammad Ashraf , Asma Ali & Shakir Ali (2004) On Lie Ideals and Generalized ( θ Φ Φ )-Derivationsin Prime Rings, Communications in Algebra, 32:8, 2977-2985, DOI: 10.1081/AGB-120039276

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On Lie Ideals and Generalized (h, })-Derivations

in Prime Rings#

Mohammad Ashraf,1,* Asma Ali,2 and Shakir Ali2

1Department of Mathematics, Faculty of Science, King Abdulaziz University,Jeddah, Saudi Arabia

2Department of Mathematics, Aligarh Muslim University, Aligarh, India

ABSTRACT

Let R be a prime ring with characteristic different from two and S a non-empty

subset of R. Suppose that y;f are endomorphisms of R. An additive mappingF : R�!R is called a generalized ðy;fÞ-derivation (resp. generalized Jordanðy;fÞ-derivation) on S if there exists a ðy;fÞ-derivation d : R�!R such that

FðxyÞ ¼ FðxÞyðyÞ þ fðxÞdðyÞ (resp. Fðx2Þ ¼ FðxÞyðxÞ þ fðxÞdðxÞ), holds for allx; y 2 S. Suppose that U is a Lie ideal of R such that u2 2 U , for all u 2 U . Inthe present paper, it is shown if y is an automorphism of R then every generalized

Jordan ðy;fÞ-derivation F on U is a generalized ðy;fÞ-derivation on U .

Key Words: Prime rings; Lie ideals; Torsion free rings; Derivations;Generalized derivations; Generalized ðy;fÞ-derivations; Generalized Jordan

ðy;fÞ-derivations.

2000 AMS Subject Classification: 16W25; 16N60; 16U80.

#Communicated by N. Gupta.

*Correspondence: Mohammad Ashraf, Department of Mathematics, Aligarh MuslimUniversity, Aligarh 202002, India; E-mail: mashraf80@hotmail.com.

COMMUNICATIONS IN ALGEBRA�

Vol. 32, No. 8, pp. 2977–2985, 2004

2977

DOI: 10.1081/AGB-120039276 0092-7872 (Print); 1532-4125 (Online)

Copyright # 2004 by Marcel Dekker, Inc. www.dekker.com

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1. INTRODUCTION

Throughout the present paper, R will denote an associative ring with centerZðRÞ. For any x; y 2 R, the symbol ½x; y� will represent the commutator xy� yx. Aring R is said to be 2-torsion free, if whenever 2x ¼ 0, with x 2 R, implies x ¼ 0.A ring R is called a prime ring if xRy ¼ ð0Þ implies that either x ¼ 0 or y ¼ 0. Anadditive subgroup U of R is said to be a Lie ideal of R if ½u; r� 2 U , for allu 2 U ; r 2 R. Let y;f be endomorphisms of R. An additive mapping f : R�!R

is called a ðy;fÞ-derivation (resp. Jordan ðy;fÞ-derivation) if fðxyÞ ¼ fðxÞyðyÞþfðxÞfðyÞ (resp. fðx2Þ ¼ fðxÞyðxÞ þ fðxÞfðxÞ), holds for all x; y 2 R. Of course a(1; 1)-derivation (resp. a Jordan (1; 1)-derivation) is a derivation (resp. a Jordanderivation) on R, where 1 is the identity mapping on R.

There has been a great deal of work concerning generalized derivation in thecontext of algebras on certain normed spaces (for reference see Hvala, 1998, wherefurther references can be found). By a generalized derivation on an algebra A oneusually means a map of the form x 7! axþ xb where a and b are fixed elements inA. We prefer to call such maps generalized inner derivations for the reason they pre-sent a generalization of the concept of inner derivations (i.e., the map of the formx 7! ax� xb). Now in a ring R, let F be a generalized inner derivation given byFðxÞ ¼ axþ xb. Notice that FðxyÞ ¼ FðxÞyþ xIbðyÞ where IbðyÞ ¼ yb� by is aninner derivation. Motivated by these observations, Hvala (1998) introduced thenotion of generalized derivation in rings. An additive mapping F : R�!R is calleda generalized derivation (resp. generalized Jordan derivation) if there exists a deriva-tion d : R�!R such that FðxyÞ ¼ FðxÞyþ xdðyÞ (resp. Fðx2Þ ¼ FðxÞxþ xdðxÞ),holds for all x; y 2 R. Hence the concept of generalized derivation covers both theconcepts of derivation and generalized inner derivation. Moreover, generalizedderivation with d ¼ 0 covers the concept of left multipliers that is an additive mapsatisfying FðxyÞ ¼ FðxÞy, for all x; y 2 R.

Inspired by the definition of ðy;fÞ-derivation, the notion of generalized deriva-tion was extended as follows: Let y, f be endomorphisms of R and let S be anonempty subset of R. An additive mapping F : R�!R is called a generalizedðy;fÞ-derivation (resp. generalized Jordan ðy;fÞ-derivation) on S if there exists aðy;fÞ-derivation d : R�!R such that FðxyÞ ¼ FðxÞyðyÞ þ fðxÞdðyÞ (resp. Fðx2Þ ¼FðxÞyðxÞ þ fðxÞdðxÞ), holds for all x; y 2 S. It is obvious to see that every generalizedðy;fÞ-derivation on R is a generalized Jordan ðy;fÞ-derivation on R but not conver-sely. Ashraf together with Wafa obtained a more general result which implies that ifR is a 2-torsion free non-commutative prime ring and if F : R�!R is a generalizedJordan derivation, then F is a generalized derivation on R (Ashraf et al., 2002). Ouraim in the present paper is to extend the above result on Lie ideals.

2. MAIN RESULT

Theorem 2.1. Let R be a 2-torsion free prime ring and U a non-commutative Lieideal of R such that u2 2 U , for all u 2 U . Suppose that y;f are endomorphisms ofR such that y is one-one, onto and d is a ðy;fÞ-derivation of R. If F :R�!R is a gen-eralized Jordan ðy;fÞ-derivation on U , then F is a generalized ðy;fÞ-derivation on U .

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Following Herstein (1969), we introduce the abbreviation xy ¼ FðxyÞ�FðxÞyðyÞ � fðxÞdðyÞ. Since F ; y;f and d are additive, for any x; y; z 2 R, we havexyþz ¼ xy þ xz and ðxþ yÞz ¼ xz þ yz.

The following lemmas are required for developing the proof of the abovetheorem:

Lemma 2.1 (Bergen et al., 1981, Lemma 4). Let R be a 2-torsion free prime ring andU a Lie ideal of R such that U 6� ZðRÞ, and aUb ¼ ð0Þ. Then a ¼ 0 or b ¼ 0.

Lemma 2.2. Let R be a 2-torsion free ring and U a Lie ideal of R such that u2 2 U ,for all u 2 U . Suppose that y, f are endomorphisms of R and d is a ðy;fÞ-deriva-tion of R. If F : R�!R is an additive mapping satisfying Fðu2Þ ¼ FðuÞyðuÞþfðuÞdðuÞ, for all u 2 U , then the following hold:

(i) Fðuvþ vuÞ ¼ FðuÞyðvÞ þ fðuÞdðvÞ þ FðvÞyðuÞ þ fðvÞdðuÞ, for all u; v 2 U .

(ii) FðuvuÞ ¼ FðuÞyðvuÞ þ fðuvÞdðuÞ þ fðuÞdðvÞyðuÞ, for all u; v 2 U .

(iii) Fðuvwþ wvuÞ ¼ FðuÞyðvwÞ þ FðwÞyðvuÞ þ fðuvÞdðwÞ þ fðwvÞdðuÞ þ fðuÞ�dðvÞyðwÞ þ fðwÞdðvÞyðuÞ, for all u; v;w 2 U .

(iv) uv½yðuÞ; yðvÞ� ¼ 0, for all u; v 2 U .

(v) uvyðwÞ½yðuÞ; yðvÞ� ¼ 0, for all u; v;w 2 U .

Proof. (i) For any u; v 2 U

Fðuvþ vuÞ ¼ Fððuþ vÞ2Þ � Fðu2Þ � Fðv2Þ¼ FðuÞyðvÞ þ fðuÞdðvÞ þ FðvÞyðuÞ þ fðvÞdðuÞ

(ii) For any u; v 2 U , uvþ vu ¼ ðuþ vÞ2 � u2 � v2, for all u; v 2 U . Replacingv by uvþ vu in (i), we get

Fðuðuvþ vuÞ þ ðuvþ vuÞuÞ ¼ FðuÞyðuvþ vuÞ þ fðuÞdðuvþ vuÞþ Fðuvþ vuÞyðuÞ þ fðuvþ vuÞdðuÞ:

Since, d : R�!R is a ðy;fÞ-derivation,

dðuvþ vuÞ ¼ dðuÞyðvÞ þ fðuÞdðvÞ þ dðvÞyðuÞ þ fðvÞdðuÞ; for all u; v 2 U

and hence

Fðuðuvþ vuÞ þ ðuvþ vuÞuÞ ¼ FðuÞyðuvÞ þ 2FðuÞyðvuÞ þ FðvÞyðu2Þþ 2fðuÞdðvÞyðuÞ þ fðvÞdðuÞyðuÞþ fðuÞdðuÞyðvÞ þ fðu2ÞdðvÞþ 2fðuvÞdðuÞ þ fðvuÞdðuÞ; for all u; v 2 U :

ð2:1Þ

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Also,

Fðuðuvþ vuÞ þ ðuvþ vuÞuÞ ¼ Fðu2vþ vu2Þ þ 2FðuvuÞ¼ FðuÞyðuvÞ þ fðuÞdðuÞyðvÞ þ FðvÞyðu2Þþ fðvÞdðuÞyðuÞ þ fðvuÞdðuÞ þ fðu2ÞdðvÞþ 2FðuvuÞ; for all u; v 2 U : ð2:2Þ

Comparing (2.1), (2.2) and using the fact that charR 6¼ 2, we obtain the requiredresult.

(iii) Replace u by uþ w in (ii), to get

Fððuþ wÞvðuþ wÞÞ ¼ Fðuþ wÞyðvuþ vwÞ þ fðuvþ wvÞdðuþ wÞþ yðuþ wÞdðvÞyðuþ wÞ

¼ FðuvuÞ þ FðwvwÞ þ FðuÞyðvuÞ þ FðwÞyðvuÞþ fðuvÞdðwÞ þ fðwvÞdðuÞ þ fðuÞdðvÞyðwÞþ fðwÞdðvÞyðuÞ; for all u; v 2 U : ð2:3Þ

On the other hand, we have

Fððuþ wÞvðuþ wÞÞ ¼FðuvuÞ þ FðwvwÞ þ Fðuvwþ wvuÞ;for all u; v 2 U : ð2:4Þ

Comparing (2.3) and (2.4), we get (iii).

(iv) For any u; v 2 U , uvþ vu and uv� vu both are in U and hence 2uv 2 U ,for all u; v 2 U . Since charR 6¼ 2, our hypothesis yields that

FððuvÞ2Þ ¼ FðuvÞyðuvÞ þ fðuvÞdðuvÞ; for all u; v 2 U :

Replacing w by 2uv in (iii), and using the fact that charR 6¼ 2, we get

FðuvðuvÞ þ uvðvuÞÞ ¼ FðuÞyðvuvÞ þ FðuvÞyðvuÞ þ fðuvÞdðuvÞþ fðuv2ÞdðuÞ þ fðuÞdðvÞyðuvÞþ fðuvÞdðvÞyðuÞ; for all u; v 2 U : ð2:5Þ

On the other hand, we have

FðuvðuvÞ þ uvðvuÞÞ ¼ FððuvÞ2Þ þ Fðuv2uÞ¼ FðuvÞyðuvÞ þ fðuvÞdðuvÞ þ FðuÞyðv2uÞþ fðuv2ÞdðuÞ þ fðuÞdðvÞyðvuÞ þ fðuvÞdðvÞyðuÞ;

for all u; v 2 U : ð2:6Þ

Comparing (2.5) and (2.6), we get the required result.

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(v) From (iii), we have

FððuvÞwðvuÞ þ ðvuÞwðuvÞÞ ¼ FðuvÞyðwvuÞ þ FðvuÞyðwuvÞ þ fðuvwÞdðvÞyðuÞþ fðuvwvÞdðuÞ þ fðvuwÞdðuÞyðvÞ þ fðvuwuÞdðvÞþ fðuvÞdðwÞyðvuÞ þ fðvuÞdðwÞyðuvÞ;

for all u; v;w 2 U : ð2:7Þ

On the other hand, we have

FððuvÞwðvuÞ þ ðvuÞwðuvÞÞ ¼ FðuðvwvÞuÞ þ FðvðuwuÞvÞ¼ FðuÞyðvwvuÞ þ fðuvwvÞdðuÞ þ fðuÞdðvwvÞyðuÞþ FðvÞyðuwuvÞ þ fðvuwuÞdðvÞ þ fðvÞdðuwuÞyðvÞ;

for all u; v;w 2 U : ð2:8Þ

Further, since 2uw 2 U , for all u;w 2 U , we find that 4uwu 2 U , for all u;w 2 U andhence

dð4uwuÞ ¼ 4dðuðwuÞÞ ¼ 4fdðuÞyðwuÞ þ fðuÞdðwÞyðuÞ þ fðuwÞdðuÞg;for all u;w 2 U :

Since, R is 2-torsion free, we have

dðuwuÞ ¼ dðuÞyðwuÞ þ fðuÞdðwÞyðuÞ þ fðuwÞdðuÞ; for all u;w 2 U :

Hence, the relation (2.8) reduces to

FððuvÞwðvuÞ þ ðvuÞwðuvÞÞ ¼ FðuÞyðvwvuÞ þ fðuvwvÞdðuÞ þ fðuÞdðvÞyðwvuÞþfðuvwÞdðvÞyðuÞ þ fðuvÞdðwÞyðvuÞ þ FðvÞyðuwuvÞþfðvuwuÞdðvÞ þ fðvÞdðuÞyðwuvÞ þfðvuwÞdðuÞyðvÞþfðvuÞdðwÞyðuvÞ; for all u; v;w 2 U : ð2:9Þ

Notice that in view of ðiÞ, xy ¼ �yx, and hence combining (2.7) and (2.9), we get therequired result.

We are now ready to prove our theorem.

Proof of Theorem 2:1. By Lemma 2.2(v), we have

uvyðwÞ�yðuÞ; yðvÞ� ¼ 0; for all u; v;w 2 U :

This yields that

y�1ðuvÞU½u; v� ¼ ð0Þ; for all u; v 2 U

and hence by Lemma 2.1, we find that for each pair u; v 2 U either y�1ðuvÞ ¼ 0 or½u; v� ¼ 0. This implies that uv ¼ 0 or ½u; v� ¼ 0, for all u; v 2 U . Now, for each

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u 2 U ; we put U1 ¼ fv 2 U j uv ¼ 0g and U2 ¼ fv 2 U j ½u; v� ¼ 0g: Clearly, both U1

and U2 are additive subgroups of U whose union is U . By Braur’s trick, we haveeither U ¼ U1 or U ¼ U2. By using similar procedure we can see that eitherU ¼ fu 2 U jU ¼ U1g or U ¼ fu 2 U jU ¼ U2g that is either uv ¼ 0; for allu; v 2 U or ½u; v� ¼ 0, for all u; v 2 U . If uv 6¼ 0, then ½u; v� ¼ 0, for all u; v 2 U , acontradiction. This completes the proof of our theorem.

Corollary 2.1. Let R be a 2-torsion free non-commutative prime ring and letF : R�!R be a generalized Jordan derivation on R. Then F is a generalizedderivation on R.

If U is a commutative Lie ideal of R, then the above result is true for y ¼ f.

Theorem 2.2. Let R be a 2-torsion free prime ring and U a non-zero commutativeLie ideal of R such that u2 2 U , for all u 2 U . Suppose that y is an automorphism ofR and d is a ðy;fÞ-derivation of R. If F : R�!R is a generalized Jordan ðy; yÞ-derivation on U , then F is a generalized ðy; yÞ-derivation on U .

Proof. Since U is a commutative Lie ideal of R i.e., ½u; v� ¼ 0, for all u; v 2 U , usingthe same arguments as used in the proof of Lemma 1.3 of Herstein (1969), we findthat U � Z. Now, by Lemma 2.2(iii), we have

Fðuvwþ wvuÞ ¼ FðuÞyðvwÞ þ FðwÞyðvuÞ þ yðuvÞdðwÞ þ yðwvÞdðuÞþ yðuÞdðvÞyðwÞ þ yðwÞdðvÞyðuÞ; for all u; v;w 2 U : ð2:10Þ

Since u2 2 U for all u 2 U , we find that uvþ vu 2 U for all u; v 2 U . This yields that2uv 2 U , for all u; v 2 U : As the ideal U is commutative, in view of Lemma 2.2ðiÞ wehave

2Fðuvwþ wvuÞ ¼ Fðð2uvÞwþ wð2uvÞÞ¼ Fð2uvÞyðwÞ þ 2yðuvÞdðwÞ þ 2FðwÞyðuvÞ þ yðwÞdð2uvÞ¼ 2fFðuvÞyðwÞ þ yðuÞyðvÞdðwÞ þ FðwÞyðuÞyðvÞ

þ yðwÞdðuÞyðvÞ þ yðwÞyðuÞdðvÞg; for all u; v;w 2 U :

This shows that

Fðuvwþ wvuÞ ¼ FðuvÞyðwÞ þ yðuÞyðvÞdðwÞ þ FðwÞyðuÞyðvÞþ yðwÞdðuÞyðvÞ þ yðwÞyðuÞdðvÞ; for all u; v;w 2 U : ð2:11Þ

Combining (2.10) and (2.11) and using the fact that uv ¼ vu, we obtain

uvyðwÞ ¼ 0; for all u; v;w 2 U : ð2:12Þ

Since y is an automorphism and w is central, we find that yðwÞ is central. But thecentral elements in a prime ring are not zero divisors and thus the equation (2.12)implies that uv ¼ 0; for all u; v 2 U : Hence we get the required result.

Following are the immediate consequences of our Theorem 2.2. &

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Corollary 2.2. Let R be a 2-torsion free prime ring and let F : R�!R be ageneralized Jordan derivation on R. Then F is a generalized derivation on R.

The following example demonstrates that R to be prime is essential in thehypothesis of the above result.

Example 2.1. Let S be a ring such that the square of each element in S is zero, butthe product of some elements in S is non-zero. Next, let

R ¼ x y

0 0

� ����� x; y 2 S

� �:

Define a map F : R ! R such that

Fx y

0 0

� �¼ 0 x

0 0

� �:

Then with d ¼ 0 and U ¼ R, it can be easily seen that Fðr2Þ ¼ FðrÞr ¼ FðrÞs ¼ 0 forall r; s 2 R, but FðrsÞ 6¼ 0 for some r; s 2 R.

If the underlying ring R is arbitrary, then we have the following:

Theorem 2.3. Let R be a 2-torsion free prime ring and U a Lie ideal of R such thatu2 2 U , for all u 2 U . Suppose that y;f are endomorphisms of R such that y isone-one, onto and d is a ðy;fÞ-derivation of R. Suppose further that U has acommutator which is not a zero divisor. If F : R�!R is a generalized Jordanðy;fÞ-derivation on U , then F is a generalized ðy;fÞ-derivation on U

Proof. Since F : R�!R is a generalized Jordan ðy;fÞ-derivation, there exists aðy;fÞ-derivation d : R�!R such that Fðu2Þ ¼ FðuÞyðuÞ þ fðuÞdðuÞ, holds for allu 2 U : Thus for any u; v 2 U if uv ¼ FðuvÞ � FðuÞyðvÞ � fðuÞdðvÞ; then by Lemma2.2 (iv), we have uv½yðuÞ; yðvÞ� ¼ 0; for all u; v 2 U . Since y is an automorphism ofR, we find that

y�1ðuvÞ½u; v� ¼ 0; for all u; v 2 U : ð2:13Þ

Let a; b be fixed elements of U such that c½a; b� ¼ 0; or ½a; b�c ¼ 0. This implies thatc ¼ 0. Hence in view of the above equation, we get y�1ðabÞ ¼ 0 i.e.,

ab ¼ 0: ð2:14Þ

Replacing u by uþ a in (2.13), we get

y�1ðuvÞ½a; v� þ y�1ðavÞ½u; v� ¼ 0; for all u; v 2 U : ð2:15Þ

Again replace v by b in (2.15), to get y�1ðubÞ½a; b� ¼ 0. Since ½a; b� is not a divisor ofzero, we have

y�1ðubÞ ¼ 0; for all u 2 U : ð2:16Þ

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Further replace v by vþ b in (2.15) and use (2.14), (2.15) and (2.16), to get

y�1ðuvÞ½a; b� þ y�1ðavÞ½u; b� ¼ 0; for all u; v 2 U : ð2:17Þ

In particular, with u ¼ a in (2.17) and using the fact that charR 6¼ 2, we havey�1ðavÞ½a; b� ¼ 0; and hence y�1ðavÞ ¼ 0 i.e.,

av ¼ 0; for all v 2 U : ð2:18Þ

Combining (2.17) and (2.18), we find that y�1ðuvÞ½a; b� ¼ 0: This implies thaty�1ðuvÞ ¼ 0 i.e., uv ¼ 0, for all u; v 2 U . Hence, F is a generalized ðy;fÞ-derivationon U .

Corollary 2.3 (Ashraf and Rehman, 2000, Theorem). Let R be a 2-torsion free ringand let F : R�!R be a generalized Jordan derivation. If R has a commutatorwhich is not a zero divisor, then F is a generalized derivation on R.

Remark 2.1. Since every ideal in a ring R is a Lie ideal of R, the conclusions of theabove theorems hold when U is assumed to be an ideal of R. Though the assumptionthat u2 2 U , for all u 2 U seems close to assuming that U is an ideal of the ring, thereexist Lie ideals with this property which are not ideals. For example, let R be any ringand U be the additive subgroup of R generated by the idempotents of R. If e is anidempotent in R, and x 2 R then it is easy to see that, u ¼ eþ ex� exe andv ¼ eþ xe� exe are idempotents. Hence, ex� xe ¼ u� v 2 U . Thus U is a Lie idealof R.

In conclusion, it is tempting to conjecture as follows:

Conjecture. Let R be a 2-torsion free prime ring and U a Lie ideal of R. Supposethat y;f are endomorphisms of R such that y is one-one, onto and d is a ðy;fÞ-derivation of R. If F : R�!R is a generalized Jordan ðy;fÞ-derivation on U , thenF is a generalized ðy;fÞ-derivation on U .

ACKNOWLEDGMENTS

The authors are greatly indebted to the referee for his=her several useful sugges-tions and valuable comments. Also, the third author gratefully acknowledges thefinancial support he received from U.G.C. India for this research.

REFERENCES

Ashraf, M., Rehman, N. (2000). On Jordan generalized derivations in rings. Math. J.Okayama Univ. 42:7–9.

Ashraf, M., Wafa, S. M., AlShammakh, A. (2002). On generalized ðy;fÞ-derivationsin rings. Internat. J. Math. Game Theo. Algebra 12:295–300.

2984 Ashraf, Ali, and Ali

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Bergen, J., Herstein, I. N., Kerr, J. W. (1981). Lie ideals and derivations of primerings. J. Algebra 71:259–267.

Herstein, I. N. (1969). Topics in Ring Theory. Chicago: Univ. Chicago Press.Hvala, B. (1998). Generalized derivations in rings. Comm. EAlgebra 26:1147–1166.

Received January 2003Revised September 2003

Lie Ideals and Generalized (h, })-Derivations in Prime Rings 2985

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