In this paper, some results concerning orthogonal generalized derivations are generalized for a nonzero ideal of a semiprime ring. These results are a generalization of results of M. Brešar and J. Vukman
in [3], which are related to a theorem of E. Posner for the product of derivations on a prime ring.</p

Albas, Emine

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On Ideals and Orthogonal Generalized Derivations of Semiprime Rings

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Mathematical Journal of OkayamaUniversityVolume 49, Issue 1 2007 Article 3JANUARY 2007On Ideals and Orthogonal GeneralizedDerivations of Semiprime RingsEmine Albas∗∗Ege UniversityCopyright c©2007 by the authors. Mathematical Journal of Okayama University is produced byThe Berkeley Electronic Press (bepress). http://escholarship.lib.okayama-u.ac.jp/mjouOn Ideals and Orthogonal GeneralizedDerivations of Semiprime RingsEmine AlbasAbstractIn this paper, some results concerning orthogonal generalized derivations are generalized fora nonzero ideal of a semiprime ring. These results are a generalization of results of M. Bresˇar andJ. Vukman in [3], which are related to a theorem of E. Posner for the product of derivations on aprime ring.KEYWORDS: derivation, generalized derivation, orthogonal generalized derivations, semiprimering, idealMath. J. Okayama Univ. 49 (2007), 53–58ON IDEALS AND ORTHOGONAL GENERALIZEDDERIVATIONS OF SEMIPRIME RINGSEmine ALBAS¸Abstract. In this paper, some results concerning orthogonal general-ized derivations are generalized for a nonzero ideal of a semiprime ring.These results are a generalization of results of M. Bres˘ar and J. Vukmanin [3], which are related to a theorem of E. Posner for the product ofderivations on a prime ring.1. IntroductionThroughout R will represent an associative ring. In [2], Bres˘ar definedthe following notion. An additive mapping D : R → R is said to be a gen-eralized derivation if there exists a derivation d : R→ R such thatD(xy) = D(x)y + xd(y) for all x, y ∈ R.By the above notion it is easily seen that the concept of a generalized deriva-tions covers both the concepts of a derivation and of a left multiplier. Thisnotion is found in P. Ribenboim [9], where some module structures of thesehigher generalized derivations were treated. Other properties of generalizedderivations were given by B. Hvala [4], T-K. Lee [5] and A. Nakajima [6],[7] and [8].Two additive maps d, g : R→ R are called orthogonal ifd(x)Rg(y) = 0 = g(y)Rd(x) for all x, y ∈ R.In [3] Bres˘ar and Vukman introduced the notion of orthogonality for twoderivations d and g on a semiprime ring, and they presented several necessaryand sufficient conditions for d and g to be orthogonal. In [10] the authorsreplaced R by a non zero ideal I of R, then they showed that some propertiesin [[3], Theorem] are also valid in this subconstruction. Finally, in [1] theauthors introduced orthogonal generalized derivations on a semiprime ringand they presented some results concerning two generalized derivations ona semiprime ring. Their results are a generalization of results of M. Bres˘arMathematics Subject Classification. 16W25, 16N60, 16D25.Key words and phrases. derivation, generalized derivation, orthogonal generalizedderivations, semiprime ring, ideal.531Albas: On Ideals and Orthogonal Generalized Derivations of SemiprimeProduced by The Berkeley Electronic Press, 200754 E. ALBAS¸and J. Vukman in [3]. In this paper, our aim is to extend their results toorthogonal generalized derivations on a nonzero ideal I of R.For a semiprime ring R and an ideal I of R, it is well-known that theleft and right annihilators of I in R coincide. Note that I ∩ `(I) = 0 (orI ∩ r(I) = 0) where `(I) and r(I) denote the left annihilator and the rightannihilator of I, respectively.Throughout this paper we assume that R is a 2-torsion free semiprimering and I is a nonzero ideal of R unless stated otherwise.2. PreliminariesIn the following, we give the notation of orthogonal generalized deriva-tions.Definition. Two generalized derivations (D, d) and (G, g) of R are calledorthogonal ifD(x)RG(y) = 0 = G(y)RD(x) for all x, y ∈ R.The following example shows that there are many pairs of generalizedderivations which are orthogonal.Example 1. Let R ={[x y0 z]: x, y, z ∈ Z, the set of integers}. Let m,p and s be fixed two nonzero elements of Z and the additive maps D, G, dand g define the following;D([x y0 z])=[0 mx + mz0 0], d([x y0 z])=[0 mx−mz0 0],G([x y0 z])=[0 pz − ys0 0]and g([x y0 z])=[0 −ys0 0]. Thenit is easy to see that d and g are derivations of R and that (D,d) and (G,g)are a generalized derivation on R such that (D,d) and (G,g) are orthogonal.Now, to obtain the main result, we need the following lemmas:Lemma 1. ([10], Lemma 1). Let R be a 2-torsion free semiprime ring, I anonzero ideal of R and a, b the elements of R. Then the following conditionsare equivalent.(i) axb = 0 for all x ∈ I.(ii) bxa = 0 for all x ∈ I.(iii) axb + bxa = 0 for all x ∈ I.Moreover, if one of the three conditions is fulfilled and `(I) = 0, then ab =ba = 0.2Mathematical Journal of Okayama University, Vol. 49 [2007], Iss. 1, Art. 3http://escholarship.lib.okayama-u.ac.jp/mjou/vol49/iss1/3ON IDEALS AND ORTHOGONAL GENERALIZED DERIVATIONS 55Lemma 2. ([10], Lemma 3). Let R be a semiprime ring and I be a nonzeroideal of R. Suppose that additive mappings F and H of R into itself satisfyF (x)IH(x) = 0 for all x ∈ I. Then F (x)IH(y) = 0 for all x, y ∈ I.3. The ResultsThe main goal of this section is to prove the following theorem, whichcorresponds to [[3], Theorem 1].Theorem 1. Let (D, d) and (G, g) be generalized derivations of R and I bea nonzero ideal of R such that `(I) = 0. Then the following conditions areequivalent.(i) (D, d) and (G, g) are orthogonal.(ii) For all x, y ∈ I, the following relations hold.(a) D(x)G(y) + G(x)D(y) = 0.(b) d(x)G(y) + g(x)D(y) = 0.(iii) D(x)G(y) = d(x)G(y) = 0 for all x, y ∈ I.(iv) D(x)G(y) = 0 for all x, y ∈ I and dG(x) = dg(x) = 0 for all x, y ∈ I.(v) (DG, dg) is a generalized derivation on I and D(x)G(y) = 0 for allx, y ∈ I.For the proof of the Theorem 1 we need the following lemmas. In all thatfollows x, y, z ∈ I and r, s, t ∈ R.Lemma 3. Let (D, d) and (G, g) be generalized derivations of R and `(I) =0. If D(I)IG(I) = 0, then D(R)RG(R) = 0.Proof. By 0 = D(x)zG(y) = G(y)zD(x) for all x, y, z ∈ I and Lemma1, we have 0 = D(x)g(r) = g(r)D(x) and by g(r)D(x) = 0, we get 0 =g(r)d(s) = d(s)g(r). Using these relations, we have D(s)xg(r) = 0 and soby 0 = D(xz)G(y), we obtain d(z)G(y) = 0. Therefore 0 = D(rx)G(sy) =D(r)xG(s)y, which shows D(r)xG(s) = 0. Replace x by r ′G(s)xD(r)r′ forsome r′ ∈ R, we have D(r)r′G(s) = 0, as desired.Moreover, we have the following:Lemma 4. Let (D, d) and (G, g) be generalized derivations of R and I anideal of R such that `(I) = 0. Then the following conditions are equivalent.(i) For any x, y ∈ I, the following relations hold.(a) D(x)G(y) + G(x)D(y) = 0.(b) d(x)G(y) + g(x)D(y) = 0.(ii) D(x)G(y) = d(x)G(y) = 0 for all x, y ∈ I.(iii) D(x)G(y) = 0 for all x, y ∈ I and dG = dg = 0 for all x, y ∈ I.(iv) (DG, dg) is a generalized derivation from I to R and D(x)G(y) = 0for all x, y ∈ I.3Albas: On Ideals and Orthogonal Generalized Derivations of SemiprimeProduced by The Berkeley Electronic Press, 200756 E. ALBAS¸Proof. (i)⇔ (ii). By (a), (b) Lemmas 1 and 2, we have 0 = D(x)zG(y) =D(x)G(y) and using this d(z)G(y) = 0. This shows (ii). And the converseis easily obtained by the relations D(x)G(y) = G(y)D(x) = 0 and Lemma 1.(ii) ⇒ (iii). By assumption, D(x)zG(y) = d(x)zg(y) = 0. Then byLemma 3, d and g are orthogonal, which shows dg = 0. Moreover, by0 = d(x)G(y) and Lemma 1, we have 0 = d(d(r)sG(y)) = d(r)sdG(y) forr, s ∈ R. Take r = G(y), we have dG(y) = 0. Since D(x)G(y) = G(y)d(x) =d(x)g(y) = 0, using Lemma 3 we obtain dG(r) = 0, this gives (iii).(iii) ⇒ (iv). By dG = dg = 0, we haveG(x)d(y) + d(x)g(y) = 0 = g(x)d(y) + d(x)g(y).Then by the proof of (i)⇒ (ii), we see that d(x)g(y) = 0 and so G(x)d(y) =0. Moreover, by 0 = D(x)G(y), we have D(x)g(z) = 0. Therefore DG(xy) =DG(x)y and thus (DG, dg = 0) is a generalized derivation from I to R.(iv)⇒ (ii). (DG, dg) is a generalized derivation if and only ifG(x)d(y) + D(x)g(y) = 0 = d(x)g(y) + g(x)d(y).So we obtain dg = 0. Furthermore by 0 = D(x)G(y), we get D(x)g(y) = 0and by the above relation, we see G(x)d(y) = 0. Therefore G(x)zd(y) = 0and by Lemma 1, we arrive at d(y)G(x) = 0. This shows (ii).Using Lemma 3 and Lemma 4, the proof of Theorem 1 is easily seen asfollows:Proof of Theorem 1. (i) ⇒ (ii), (iii), (iv) and (v) are clear by [[1], Theorem1]. Since (ii), (iii), (iv) and (v) are equivalent by Lemma 4, we assume (iii).This implies that 0 = (D(x)z + xd(z))G(y) = D(x)zG(y). Then we haveD(I)IG(I) = 0. Thus by Lemma 3, we have Theorem 1 (iii) ⇒ (i).Remark 1. If (DG, dg) is a generalized derivations on I and `(I) = 0 then(DG, dg) is a generalized derivations on R.Proof. It is easily seen that (DG, dg) is a generalized derivations on I ifand only ifG(x)d(y) + D(x)g(y) = 0, d(x)g(y) + g(x)d(y) = 0.Then by the second relation, we have d and g are orthogonal. By the firstrelation 0 = G(x)d(y) + D(x)g(y), we get 0 = G(x)zd(y) + D(x)zg(y).Hence replacing z by g(y)z in this relation and using the orthogonality4Mathematical Journal of Okayama University, Vol. 49 [2007], Iss. 1, Art. 3http://escholarship.lib.okayama-u.ac.jp/mjou/vol49/iss1/3ON IDEALS AND ORTHOGONAL GENERALIZED DERIVATIONS 57of the derivations d and g, we obtain 0 = D(x)g(y)zg(y) which impliesthat D(x)g(y) = G(x)d(y) = 0. Moreover by 0 = D(x)g(yr), we get0 = D(x)g(r) for all r ∈ R. Using this relation we have D(s)xg(r) = 0and similarly we can see that D(s)g(r) = G(s)d(r) = 0. Thus we obtainDG(rs) = DG(r)s for all r, s ∈ R which completes the proof.Theorem 2. Let (D, d) and (G, g) be generalized derivations of R and`(I) = 0. Then the following conditions are equivalent.(i) (DG, dg) is a generalized derivation on I.(ii) (GD, gd) is a generalized derivation on I.(iii) D and g are orthogonal, and G and d are orthogonal.The proof of the Theorem 2 is clear by Remark 1 and [[1], Theorem 2].Corollary 1. Let (D, d) be a generalized derivations of R and `(I) = 0. If(D2, d2) is a generalized derivation on I, then d = 0.Proof. The fact that (D2, d2) is a generalized derivation on I is impliesthat d and d are orthogonal. Therefore we get d = 0 by the semiprimenessof R.Corollary 2. Let (D, d) be a generalized derivations of R and `(I) = 0. IfD(x)D(y) = 0 for all x, y ∈ I, then D = d = 0.Proof. By the hypothesis we have 0 = D(x)D(yz) = D(x)D(y)z +D(x)yd(z) = D(x)yd(z) for all x, y, z ∈ I. In particular, we see thatd(z)D(x) = 0 for all x, z ∈ I by Lemma 1. Replacing x by xy in the lastrelation we get 0 = d(z)D(x)y + d(z)xd(y) = d(z)xd(y) for all x, y, z ∈ I.By [[10], Lemma 2, (a) and (b)], we obtain d(s)Rd(r) = 0 for all s, r ∈ R.In particular d(s)Rd(s) = 0 for all s ∈ R. Thus d = 0 by the semiprime-ness of R. Then we have 0 = D(xz)D(y) = D(x)zD(y) for all x, y, z ∈ I.By Lemma 3, we arrive at D(r)RD(s) = 0 for all r, s ∈ R. In particular,D(r)RD(r) = 0 for all r ∈ R which implies D = 0, as desired.Acknowledgement. The author would like to thank the referee forhis/her valuable suggestions which simplify some arguments of this paper.References[1] N. Argac¸, A. Nakajima and E. Albas¸: On Orthogonal generalized derivations ofsemiprime rings, Turk J. Math. 28 (2004), 185-194.[2] M. Bres˘ar: On the distance of the compositions of two derivations to the generalizedderivations, Glasgow Math. J. 33 (1991), 80-93.5Albas: On Ideals and Orthogonal Generalized Derivations of SemiprimeProduced by The Berkeley Electronic Press, 200758 E. ALBAS¸[3] M. Bres˘ar and J. Vukman: Orthogonal derivation and extension of a theorem ofPosner, Radovi Matematic˘ki 5 (1989), 237-246.[4] B. Hvala: Generalized derivations in prime rings, Comm. Alg. 26(4) (1998), 1147-1166.[5] T-K. Lee: Generalized derivations of left faithful rings, Comm. Alg. 27(8) (1999),4057-4073.[6] A. Nakajima: On categorical properties of generalized derivations, Scientiae Mathe-maticae 2 (1999), 345-352.[7] A. Nakajima: Generalized Jordan derivations, International Symposium on Ring Tho-ery , 2000, Birkha¨user.[8] A. Nakajima: On generalized higher derivations, Turkish J. Math. 24(2000), 295-311.[9] P. Ribenboim: High order derivations of modules, Portgaliae Math. 39(1980), 381-397.[10] M. S¸. Yenigu¨l and N. Argac¸: On ideals and orthogonal derivations, J. of SouthwestChina Normal Univ. 20. (1995), 137-140.Ege University,Department of Mathematics,Science Faculty35100 Bornova, Izmir-Turkeye-mail address: emine.albas@ege.edu.tr, eminealbas@yahoo.com(Received July 29, 2005 )6Mathematical Journal of Okayama University, Vol. 49 [2007], Iss. 1, Art. 3http://escholarship.lib.okayama-u.ac.jp/mjou/vol49/iss1/3

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