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12
Left Multiplicative Generalized Derivations on
Lie Ideals in Prime Rings
Dr. C. Jaya Subba Reddy, S. Mallikarjuna Rao, K. Hemavathi
Department of Mathematics, S.V. University, Tirupati, Andhra Pradesh, India
International Journal of Research in Mathematics & Computation
Volume 2, Issue 2, July-December, 2014, pp. 12-18
ISSN Online: 2348-1528, Print: 2348-151X, DOA : 03102014
© IASTER 2014, www.iaster.com
ABSTRACT
Let be a ring. A map is called a left multiplicative generalized derivation, if
) is fulfilled for all in ,where is any map (not
necessarily derivation or additive map). In this paper we prove that Let be a 2-torsion free prime
ring and be a nonzero square closed Lie ideal of If is a left multiplicative generalized
derivation associated with the map such that
1) , then and , for all .
2) , then , and , for all .
3) , then and , for all .
4) , then and .
Keywords: Prime ring, Derivation, Generalized derivation, Multiplicative generalized derivation,
Left multiplicative generalized derivation, Ideals, Lie ideals.
1. INTRODUCTION
Ashraf and Rehman[3], proved that if is a prime ring with a nonzero ideal and is a derivation of
such that either for all or for all ,
then is commutative. Being inspired by this result, recently Ashraf et al. [4] have studied the
situation obtained by replacing derivation with a generalized derivation . Moreover, Ali et
al.[5]proved similar results by considering a square closed Lie ideal of a prime ring with
generalized derivation Motivated by these results Ali et al.[2] considered the similar situation with
multiplicative generalized derivation on a square closed Lie ideal of a prime ring. Being inspired by
this result, we consider the similar situation with left multiplicative generalized derivation on a square
closed Lie ideal of a prime ring.
Preliminaries: Throughout this paper will denote an associative ring with centre For
any in the symbol is called commutator. We recall that is prime if for
any in implies or . Center is defined as
International Journal of Research in Mathematics & Computation
Volume
-
2
, Issue
-
2
,
July
-
December
, 201
4
, www.iaster.com
ISSN
(O) 2348-1528
(P) 2348
-
151X
13
. An additive map is called a derivation of if
for all in An additive map is called a generalized
derivation if there exists a derivation such that for all
A map (not necessarily additive) is called a multiplicative generalized
derivation if for all where is any map (not necessarily
derivation or additive map). A map (not necessarily additive) is called left multiplicative
generalized derivation if for all where is any map (not
necessarily derivation or additive map). And an additive subgroup of is said to be a Lie ideal of
if for all and . is said to be a square closed Lie ideal of if
for all . Moreover if is a square closed Lie ideal of then for all
We use the following basic identities:
and for all .
2. MAIN RESULTS
Lemma 1: Let be a 2- torsion free prime ring and be a Lie ideal of . If then
.
Lemma 2: If is a Lie ideal of a 2-torsion free prime ring and are such
that , then or .
Lemma 3: Let be a 2-torsion free prime ring and U be Lie ideal of . If is a commutativeLie
ideal of R,then .
Theorem 1:Let be a 2-torsion free prime ring and be a nonzero square closed Lie ideal of If
admits a left multiplicative generalized derivation associated with the map such
that for all , then and for all
Proof: Assume first that (1)
We replace by in equation (1) we get
From equation (1) we get
Since R is 2- torsion free ring (2)
We replace by in equation (2) we get
International Journal of Research in Mathematics & Computation
Volume
-
2
, Issue
-
2
,
July
-
December
, 201
4
, www.iaster.com
ISSN
(O) 2348-1528
(P) 2348
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151X
14
Since is prime ring and is a nonzero Lie ideal of
(3)
We replace by in equation (3) we get
Since is prime ring is nonzero ideal then
(4)
Now (5)
(6)
We replace by in equation (6) we get
From equation (6) we get
Using the primeness of we get
Hence the theorem.
Theorem 2: Let be a 2- torsion free prime ring and be a nonzero square closed Lie ideal of
If admits a left multiplicative generalized derivation associated with the map
such that for all , then , and
.
Proof: Suppose
We have (7)
We replace by in equation (7) we get
From equation (7) we get above expression
Using the fact 2-torsion free ring we get
International Journal of Research in Mathematics & Computation
Volume
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2
, Issue
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2
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July
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ISSN
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(P) 2348
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(8)
Substitute by in the equation (8) we obtain
(9)
We replace by in equation (9) we get
This implies
Taking by the above expression becomes
From equation (9) we get
We replace by and using the 2- torsion freeness of we get the above expression
By Lemma2 we obtain
(10)
We replace by in equation (8) we get
From equation (10) the above expression becomes
(11)We replace by in equation (10) we get
Using 2-torsion freeness
From equation (11) the above expression becomes
From Lemma 2
Again using Lemma3 we get contradiction
We must have
Then our assumption become
By Theorem 1 we get and
In similar manner we can prove our conclusion when .
Theorem 3: Let be a 2-torsion free prime ring be a nonzero square closed Lie ideal of If
admits a left multiplicative generalized derivation associated with the map
such that , then and .
Proof: First consider case (12)
We replace by in equation (12) we get
International Journal of Research in Mathematics & Computation
Volume
-
2
, Issue
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2
,
July
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December
, 201
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(P) 2348
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From equation (12) the above expression becomes
Since is 2-torsion free ring
(13)
Right multiplying equation (13) by we get
From equation (12) we have
(14)
We replace by in equation (14) we get
From equation (14) the above expression becomes
Since is a prime ring and is a nonzero Lie ideal of
(15)
We replace in equation (15) we get
From equation (15) the above expression becomes
Prime ring , is nonzero ideal of
(16)
We replacing by in equation (16) we get
From equation (16) the expression becomes
By primeness of , is nonzero,
(17)
Thus for any we have
(18)
On replacing by in equation (12)
International Journal of Research in Mathematics & Computation
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Since is 2-torsion free ring we obtain
(19)
Left multiplying in equation (12) we obtain
(20)
Subtract equation (19)and (20) we get
(21)
Substitute by in (21) we obtain
Using 2-torsion free ring we have
It follows
If , lemma2 gives and the same condition is obtained if
.
In a similar manner, we can prove that the same conclusion holds for
.
Hence the theorem.
Theorem 4: Let be a 2-torsion free prime ring and be a nonzero square closed Lie ideal of . If
admits a left multiplicative generalized derivation associated with the map such
that , then and .
Proof: Suppose on contrary that
Assume that (22)
We replace by in equation (22) we obtain
The above expression is 2- torsion free ring
(23)
From equations (22) & (23), we obtain
(24)
Right multiplying equation (24) with we get
From equation (22) we get (25)
We replace by in equation (25) we obtain
International Journal of Research in Mathematics & Computation
Volume
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2
, Issue
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2
,
July
-
December
, 201
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ISSN
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(P) 2348
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151X
18
By lemma2 we have for each
Since
Now replacing by in equation (22)
Since we obtain
(26)
Left multiplying equation (22) by we get
(27)
Subtract equations (26) & (27) we get
Then same argument as given in the proof of Theorem 2 we have a contradiction. We
must have .
In a similar manner, we can prove that the same conclusion holds for
.
There by the proof of the theorem is completed.
REFERENCES
[1] Ali,A., Ali,S., Rani,R.: On Lie Ideals and Generalized Derivations of Prime Rings. East-West
J.Math.7(1) 93-98(2005).
[2] Asharf,M.,Ali,A., Ali,S.: Some Commutativity Theorems for Rings with Generalized
Derivations. Southeast Asian Bull. Math.31,415-421 (2007).
[3] Asharf,M., Rehaman,N.: On Derivations and Commutativity in Prime Rings. East-West
J.math.3(1) 87-91(2001).
[4] Bergen,J.,Heristein,I.N.,Kerr,J.W.: Lie Ideals and Derivations of Prime Rings. J. Algebra
71,259-267(1981).
[5] Daif. M.N., Bell, H.E.: Remarks on Derivations on Semi Prime Rings.
Internat.J.Math.Math.Sci.15(1),205-206(1992).
[6] Rehman,N.: On Commutativity of Prime Rings with Generalized Derivations.
J.OkayamaUniv.44, 43-49(2002).
[7] Shakir Ali, BasudebDharaNadeem, Ahamad Dar, Abdul Nadim Khan: On Lie Ideals with
Multiplicative(Generalized)-Derivations in Prime and Semiprime Rings, Beitra Algebra
Geom.(2013).
