Volume 2009, Article ID 696737,6pages doi:10.1155/2009/696737
Research Article
Dependent Elements of Derivations on Semiprime Rings
Faisal Ali and Muhammad Anwar Chaudhry
Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan, Pakistan
Correspondence should be addressed to Muhammad Anwar Chaudhry,[email protected] Received 15 October 2008; Accepted 19 February 2009
Recommended by Nils-Peter Skoruppa
We characterize dependent elements of a commuting derivationdon a semiprime ringRand investigate a decomposition ofRusing dependent elements ofd. We show that there exist ideals UandV ofRsuch thatU⊕V is an essential ideal ofR,U∩V {0},d0 onU,dV⊆V, andd acts freely onV.
Copyrightq2009 F. Ali and M. A. Chaudhry. 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.
1. Introduction and Preliminaries
Murray and von Neumann1and von Neumann2introduced the notion of free action on abelian von Neumann algebras and used it for construction of certain factors. Kallman 3 generalized the notion of free action of automorphisms to von Neumann algebras, not necessarily abelian, by using implicitly the dependent elements of an automorphism.
Dependent elements of automorphisms were later studied by Choda et al.4in the context of C∗-algebras. Several other authors have studied dependent elements of automorphisms in the context of operator algebrassee5,6and references therein. A brief account of dependent elements inW∗-algebras has also appeared in the book of Str˘atil˘a7.
It is well known that all C∗ and von Neumann algebras are semiprime rings; in particular a von Neumann algebra is prime if and only if its centre consists of the scalar multiples of identity8. Thus a natural extension of the notion of a dependent element of mappings on aC∗-algebra or von Neumann algebras is the study of this notion in the context of semiprime rings and prime rings.
Laradji and Thaheem9initiated the study of dependent elements of endomorphisms of semiprime rings and generalized a number of results of4for semiprime rings. Recently,
Vukman and Kosi-Ulbl 10and Vukman 11, 12 have made further study of dependent elements of some mappings on prime and semiprime rings.
On one hand, motivated by the work of Laradji and Thaheem9, Vukman and Kosi- Ulbl 10, and Vukman 11, 12 on dependent elements of mappings of semiprime rings and on the other hand by the work done by various researchers on commuting derivations on prime and semiprime rings, we investigate some properties, not already investigated, of dependent elements of commuting derivations on semiprime rings. We show that the dependent elements of a commuting derivation of a semiprime ring are central and form a commutative semiprime subring ofR. We also show that for a commuting derivationdon a semiprime ringR, there exist idealsUandV ofRsuch thatU⊕V is an essential ideal of R,U∩V {0},d0 onU,dV⊆V, and zero is the only dependent element ofd\V, the restriction ofdonV; that is,dacts freely onV.
Throughout,R will represent an associative ring with centreZR. The commutator xy−yx will be denoted by x, y. We will use the basic commutator identitiesxy, z x, zyxy, zandx, yz yx, z x, yz. Recall that a ringRis semiprime ifaRa0 impliesa0 and is prime ifaRb0 impliesa0 orb0. An additive mappingd:R → R is called a derivation onRifdxy dxyxdyfor allx, y∈R. It is called commuting if dx, x 0 for allx∈R. Leta∈R, then the mappingd:R → Rgiven bydx a, xis a derivation onR. It is called inner derivation onR.
We call an element elementa∈Ra dependent element of a derivationd:R → Rif dxa a, xaholds for allx∈R. Following3, a derivationd:R → Ris said to act freely onR or a free actionin case zero is the only dependent element of d. It is known that a semiprime ringRhas no central nilpotent elements. We will use this fact without any specific reference. For a derivationd:R → R,Dddenotes the collection of all dependent elements ofd.
It is known that the left and right annihilators of an ideal Uof a semiprime ringR coincide. It will be denoted by AnnU. It is also known thatU∩AnnU {0}andU⊕ AnnUis an essential ideal ofR. We will use these facts without any further reference.
We will use the following result in the sequel.
Theorem 1.1see8, Corollary 3.2. Ifdis a commuting inner derivation on a semiprime ringR, thend0.
2. Results
We now prove our results.
Theorem 2.1. Letdbe a commuting derivation of a semiprimeringR. Thena∈Ddif and only if a∈ZRanddxa0 for allx∈R.
Proof. Leta∈Dd. Then
dxa a, xa ∀x∈R. 2.1 Replacingxbyxyin2.1, we getdxya a, xya. That is,
dxyaxdyaxa, ya a, xya ∀x, y∈R. 2.2
From2.1and2.2, we get
dxya a, xya ∀x, y∈R. 2.3
Multiplying2.3byzon the right, we get
dxyaz a, xyaz. 2.4
Replacingybyyzin2.3, we get
dxyza a, xyza. 2.5
Subtracting2.5from2.4, we getdxyaz−za a, xyaz−za, which implies
dxya, z a, xya, z. 2.6
Multiplying2.6byxon the left, we get
xdxya, z xa, xya, z. 2.7
Replacingybyxyin2.6, we get
dxxya, z a, xxya, z. 2.8
Subtracting2.7from2.8, we get
dx, xya, z a, x, xya, z. 2.9
Sincedis commuting, therefore from2.9we get
a, x, xya, z 0. 2.10
From2.10, we get
a, x, xya, zz0. 2.11
Replacing y by yz in 2.10 and then subtracting the result from 2.11, we get a, x, xya, z, z 0, which impliesa, x, xya, x, x 0. Using semiprimeness ofR, from the last relation we get
a, x, x 0 ∀x∈R. 2.12
Thus inner derivationψ :R → Rdefined byψx a, xis commuting. Henceψx 0 by Theorem 1.1, which impliesa, x 0. Thusa∈ZR. Further from2.1, we getdxa0.
Conversely, leta∈ZRand letdxa0. Thendxa0 a, xa. Soa∈Dd. This completes the proof.
Corollary 2.2. LetRbe a semiprime ring and letdbe a commuting derivation ofR. Leta∈Dd, thenda 0.
Proof. Sincea∈Dd, therefore
dxa0 ∀x∈R. 2.13
Replacingxbydxin2.13, we get
d2xa0 ∀x∈R. 2.14
From2.13, we getddxa d0 0, which impliesd2xadxda 0. Using2.14, from the last relation we get
dxda 0. 2.15
Replacingxbyaxin2.15and using2.15, we get 0daxda daxdaadxda daxda. Thusdaxda 0 for allx∈R. Using semiprimeness ofR, from the last equation we getda 0.
Corollary 2.3. LetRbe a semiprime ring and letdbe a commuting derivation ofR. ThenDdis a commutative semiprime subring ofR.
Proof. Leta, b ∈ Dd. Then byTheorem 2.1a, b ∈ ZR,dxa 0, anddxb 0 for all x ∈R. Obviouslya−b ∈ZRanddxab 0. Soa−bandab ∈ Dd. Sincea, b ∈ZR, soab ba. ThusDdis a commutative subring ofR. To show semiprimeness ofDd, we consideraDda 0,a∈Dd. Thenaxa 0 for allx ∈Dd. In particulara3 0, which impliesa 0 becauseRhas no central nilpotents. ThusDdis a commutative semiprime subring of ring.
Corollary 2.4. LetRbe a commutative semiprime ring and letdbe a derivation ofR. ThenDdis an ideal ofR.
Proof. Since R is commutative, sodis commuting. Leta, b ∈ Dd. Then byCorollary 2.3, a−b∈Dd. Leta∈Ddand letr∈R. Thendxa0 for allx∈R. Thusdxar0. Since arra, sodxardxra0 for allx∈R. Hencearra∈Dd. ThusDdis an ideal of R.
Remark 2.5. iIfRis a semiprime ring andUan ideal ofR, then it is easy to verify thatUis a semiprime subring ofRandZU⊆ZR.
iiIfdis a commuting derivation onRanda∈Dd, then byTheorem 2.1,dxa0 for allx∈R. This implies 0dxyadxyaxdyadxya, which givesdxya0.
Thusadxyadx 0 for allx, y∈R, which by semiprimeness ofRimpliesadx 0.
Theorem 2.6. LetRbe a semiprime ring and letdbe a commuting derivation onR. Then there exist idealsUandVofRsuch that
aU⊕Vis an essential ideal ofRandU∩V {0}, bd0 onUanddV⊆V,
cDd\V {0}, whered\V is restriction ofdonV. That is,dacts freely onV.
Proof. aLetUbe the ideal ofRgenerated byDd. LetV AnnU. ThenV is an ideal of R,U⊕V is an essential ideal ofR, andU∩V {0}.
bByCorollary 2.2andTheorem 2.1,da 0 anddxa 0 for allx ∈ Randa ∈ Dd. ByRemark 2.5iiadx 0. Thusdax daxadx 0,dxa dxaxda 0, anddxay dxayxdayxady 0 fora∈Ddandx, y∈R. Henced0 onU.
Letv ∈V AnnU. Thusva 0 for alla∈U. Sodva d0 0, which implies dvavda 0. Thusdva 0 becaused 0 onU. So,dv ∈ AnnU V. Hence dV⊆V.
cSinceV is an ideal ofR, so byRemark 2.5iV is a semiprime subring ofRand ZV ⊆ ZR. SincedV ⊆ V, sod\V is a derivation on V. Let c ∈ V be a dependent element ofd\V, so byTheorem 2.1andCorollary 2.2 c ∈ZV ⊆ ZR,d\Vc 0, and d\Vvccd\Vv 0. Letx∈R, soxv∈V. Thusd\Vxvc0, which impliesdxvc0.
That is,dxvcxdvc0, which impliesdxvc0. SinceVis an ideal ofR, socvdx∈V anddxc∈V for allx ∈R. Replacingvbycvdxindxvc0, we getdxcvdxc0.
Using semiprimeness ofV, we getdxc 0. Sincec ∈ ZRanddxc 0 for allx ∈ R, thereforec∈Dd⊆U. Soc∈Uandc∈V AnnU. Thusc 0. HenceDd\V {0}.
That is,dacts freely onV.
Since every derivationdon a commutative ring is a commuting derivation andDd is an ideal of R by Corollary 2.4, therefore in case of a commutative ring U Ddand V AnnDd. Thus we have the following corollary.
Corollary 2.7. LetRbe a commutative semiprime ring and letdbe a derivation onR. Then there exist idealsUDdandV AnnDdofRsuch that
aU⊕Vis an essential ideal ofRandU∩V {0}, bd0 onU,dV⊆V,
cDd\V {0}, whered\V is restriction ofdonV. That is,dacts freely onV.
The authors are thankful to the referee for suggesting another proof ofTheorem 2.6 with different idealsV andU. The idealVis generated bydRandUAnnRV. The proof suggested by the referee is based on a theorem of Chuang and Lee13. The statement of the said theorem and the proof ofTheorem 2.6as suggested by the referee are given below.
Theorem 2.8see13. LetRbe a semiprime ring with a derivationdand letλbe a left ideal ofR.
Suppose thatdx, x∈ZRfor allx∈λ, whereZRdenotes the center ofR. Thenλ, RdR 0.
Now we give the proof ofTheorem 2.6as suggested by the referee.
Proof. By Theorem 2.8, R, RdR 0. LetV be the ideal of R generated bydRand let UAnnRV. Clearly,R, R⊆U,U∩V 0,U⊕V is essential inR, anddV⊆V ⊆ZR.
SincedU⊆U∩V, we havedU 0. Letcbe a dependent element ofd\V. Letv∈V. Then dvc c, vc∈UV 0. ThusdU⊕VcdVc 0. SinceU⊕V is an essential ideal of the semiprime ringR, we havedRc0, implying thatc∈U. Soc∈U∩V 0, as asserted.
This proves the theorem.
Acknowledgment
The authors are grateful for the support provided by Bahauddin Zakariya University, Multan, Pakistan.
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