NOTES ON GENERALIZED DERIVATIONS OF -PRIME RINGS

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Miskolc Mathematical Notes HU e-issn 1787-2413 Vol. 15 (2014), No. 1, pp. 117 123 NOTES ON GENERALIZED DERIVATIONS OF -PRIME RINGS EMINE KOÇ AND NADEEM UR REHMAN Received 24 September, 2013 Abstract. Let R be a -prime ring with characteristic different from two and U 0 be a square closed -Lie ideal of R. An additive mapping F W R! R is called an generalized derivation if there exits a derivation d W R! R such that F.xy/ D F.x/y C xd.y/. In the present paper, it is shown that U Z if R is a -prime ring which admits a generalized derivation satisfying several conditions that are associated with a derivation commuting with. 2010 Mathematics Subject Classification: 16W25; 16N60; 16U80 Keywords: prime rings, derivations, generalized derivations, Lie ideals 1. INTRODUCTION, ETC Let R will be an associative ring with center Z: For any x;y 2 R; denote the commutator xy yx by Œx;y and the anti-commutator xy C yx by x ı y. Recall that a ring R is prime if xry D 0 implies x D 0 or y D 0: An additive mapping W R! R is called an involution if.xy/ D y x and.x / D x for all x;y 2 R: A ring equipped with an involution is called a ring with involution or ring. A ring with an involution is said to prime if xry D xry D 0 or xry D x Ry D 0 implies that x D 0 or y D 0: Every prime ring with an involution is prime but the converse need not hold general. An example due to Oukhtite [6] justifies the above statement. Suppose that R is a prime ring, S D R R o where R o is the opposite ring of R: Define involution on S as.x;y/ D.y;x/: S is prime, but not prime. This example shows that every prime ring can be injected in a prime ring and from this point of view prime rings constitute a more general class of prime rings. In all that follows the symbol S a.r/; first introduced by Oukhtite, will denote the set of symmetric and skew symmetric elements of R; i.e. S a.r/ D fx 2 R j x D xg: An additive subgroup U of R is said to be a Lie ideal of R if ŒU;R U: A Lie ideal is said to be a -Lie ideal if U D U: If U is a Lie (resp. -Lie) ideal of R; then U is called a square closed Lie (resp. -Lie) ideal of R if x 2 2 U for all x 2 U. c 2014 Miskolc University Press

118 EMINE KOÇ AND NADEEM UR REHMAN An additive mapping d W R! R is called a derivation if d.xy/ D d.x/y C xd.y/ holds for all x;y 2 R: In particular, for fixed a 2 R; the mapping I a W R! R given by I a.x/ D Œa;x is a derivation which is said to be an inner derivation. An additive function F W R! R is called a generalized inner derivation if F.x/ D ax C xb for fixed a;b 2 R. For such mapping F, it is easy to see that F.xy/ D F.x/y C xœy;b D F.x/y C xi b.y/; for all x;y 2 R: This observation leads to the following definition, an additive mapping F W R! R is called a generalized derivation associated with a derivation d W R! R if F.xy/ D F.x/y C xd.y/ holds for all x;y 2 R. Familiar examples of generalized derivations are derivations and generalized inner derivations, and the latter includes left multipliers. Since the sum of two generalized derivation is a generalized derivation, every map of the form F.x/ D cx Cd.x/, where c is a fixed element of R and d a derivation of R is a generalized derivation, and if R has multiplicative identity, then all generalized derivations have this form. Over the past thirty years, there has been an ongoing interest concerning the relationship between the commutativity of a prime ring R and the behavior of a special mapping on that ring. Recently, some well-known results concerning prime rings have been proved for prime rings by Oukhtite et al. (see [4 8], where further references can be found). In the year 2005, Ashraf et al. [1] proved some commutativity theorems for prime rings. Recently the first author with Al-Omary [3] obtained the commutativity of -prime ring R admitting generalized derivations satisfying several conditions. Motivated by the above results, in this paper we shall discuss the situation when a -prime ring R which admits a generalized derivation F associated with a derivation d satisfying any one of the following properties:.i/ d.u/of.v/ D Œu;v ;.ii/ Œd.u/;F.v/ D.u ı v/.iii/ d.u/f.v/ D Œu;v.iv/ d.u/f.v/ D.u ı v/.v/ Œd.u/;F.v/ D uv and.vi/.d.u/ ı F.v// D uv. 2.1. Preliminary considerations 2. RESULTS The followings are some useful identities which hold for every x;y; 2 R. We will use them in the proof of our theorems. Œx;y D yœx; C Œx;y Œxy; D Œx; y C xœy; xo.y / D.xoy/ yœx; D y.xo / C Œx;y.xy/ ı D x.y ı / Œx; y D.x ı /y C xœy; : We begin our discussion with the following results.

NOTES ON GENERALIZED DERIVATIONS OF -PRIME RINGS 119 Lemma 1 ([7, Lemma 4]). Let R be a prime ring with characteristic not two, U be a nonzero Lie ideal of R and a;b 2 R: If aub D aub D 0; then a D 0 or b D 0 or U Z: Lemma 2 ([2, Lemma 2.7]). Let R be a prime ring with characteristic not two, U be a nonzero Lie ideal of R. If a 2 R such that Œa;U Z; then either a 2 Z or U Z: The following Lemma is immediate consequences of Lemma 2. Lemma 3. Let R be a prime ring with characteristic not two and U be a nonzero Lie ideal of R. Suppose that ŒU;U Z; then U Z: Lemma 4 ([5, Lemma 2.4]). Let R be a prime ring with characteristic not two, d be a nonzero derivation of R which commutes with and U be a nonzero Lie ideal of R: If d.u / Z; then U Z: Lemma 5 ([5, Lemma 2.5]). Let R be a prime ring with characteristic not two, d be a nonzero derivation of R which commutes with and U be a nonzero Lie ideal of R: If a 2 R and ad.u / D 0.d.U /a D 0/; then a D 0 or U Z: Lemma 6 ([5, Theorem 1.1]). Let R be a prime ring with characteristic not two, d be a nonzero derivation of R which commutes with and U be a nonzero Lie ideal of R: If d 2.U / D 0; then U Z: 2.2. Something else Theorem 1. Let R be a prime ring with characteristic not two and U be a nonzero square closed Lie ideal of R: Suppose that R admits a generalized derivation F associated with nonzero derivation d which commutes with such that.i/.d.u/ ı F.v// D Œu;v for all u;v 2 U; or.ii/ Œd.u/;F.v/ D.u ı v/ for all u;v 2 U; or.iii/ Œd.u/;F.v/ D uv for all u;v 2 U; or.iv/ d.u/ ı F.v/ D uv for all u;v 2 U; or If F D 0 or d 0, then U Z: Proof. Suppose on the contrary that U 6 Z. Write L D ŒU;U, then it is easy to show that L is a Lie ideal and d.l/ U. Moreover, since U 6 Z, then L 6 Z by Lemma 3..i/ If F D 0; then by Lemma 3, we get U Z, a contradiction. Henceforth, we shall assume that d 6D 0. We have.d.u/ ı F.v// D Œu;v for all u;v 2 U: (2.1) Replacing v by 2vw in (2.1) and using (2.1), and the fact that charr 2; we conclude that F.v/Œd.u/;w C v.d.u/ ı d.w// C Œd.u/;v d.w/ D v Œu;w ; (2.2)

120 EMINE KOÇ AND NADEEM UR REHMAN for all u;v;w 2 U. Now, choose w D d.u/ for all u 2 L in the above expression we find that v d.u/ ı d 2.u/ C Œd.u/;v d 2.u/ D v Œu;d.u/ : (2.3) Again, replace v by 2vw in (2.3) and use (2.3), to get Œd.u/;v Ud 2.u/ D 0 for all v 2 U;u 2 L: (2.4) Since U is a nonzero square closed Lie ideal of R; L is a nonzero square closed Lie ideal of R; too. Hence using d D d, we get Œd.u/;v U.d 2.u// D 0 for all v 2 U;u 2 L \ S a.r/: Thus, by Lemma 1, we get either Œd.u/;v D 0, for all v 2 U or d 2.u/ D 0 for each u 2 L \ S a.r/: Let u 2 L; as u C u ;u u 2 L \ S a.r/ and Œd.u u /;v D 0, for all v 2 U or d 2.uu / D 0: Hence, we have Œd.u/;v D 0 or d 2.u/ D 0; for all v 2 U;u 2 L: We obtain that L is the set theoretic union of two its proper subgroups viz. A D fu 2 L j Œd.u/;v D 0g and B D fu 2 L j d 2.u/ D 0g: But a group cannot be the set-theoretic union of two proper subgroups, hence A D L or B D L. If A D L, then Œd.u/;U D 0 for all u 2 L; and hence by Lemma 2 we find that d.u/ 2 Z for all u 2 L. Thus, by Lemma 4 and Lemma 3, we have U Z, a contradiction. On the other hand, if B D L, then d 2.L/ D 0 that is, L Z by Lemma 6, and so again using Lemma 3, we get the required result..ii/ If F D 0; then u ı v D 0; for all u;v 2 U: Replacing v by 2vw; w 2 U in the last equation, we get U Œu;w D 0; for all u;w 2 U: Hence we arrive at vu Œu;w D 0 for all u;v;w 2 U: (2.5) Since U is a nonzero Lie ideal of R yields that v U Œu;w D 0 for all u;v;w 2 U: Hence, we have vu Œu;w D v U Œu;w D 0 for all u;w;v 2 U: By Lemma 1, we get either Œu;w D 0; for all u;w 2 U or v D 0 for each v 2 U: And so, Œu;w D 0; for all u;w 2 U: By Lemma 3, we obtain U Z: Therefore, we shall assume that d 0: We have Œd.u/;F.v/ D u ı v for all u;v 2 U: (2.6) Replacing v by 2vw in (2.6 and using (2.6), we conclude that F.v/Œd.u/;w C vœd.u/;d.w/ C Œd.u/;v d.w/ D v Œu;w ; (2.7) for all u;v;w 2 U: For any u 2 L, replace w by d.u/ in (2.7), to get vœd.u/;d 2.u/ C Œd.u/;v d 2.u/ D v Œu;d.u/ ; (2.8)

NOTES ON GENERALIZED DERIVATIONS OF -PRIME RINGS 121 for all v 2 U;u 2 L. Now, replacing v by 2vw in (2.8) and using (2.8), we see that Œd.u/;v Ud 2.u/ D 0 for all v 2 U;u 2 L: Notice that the arguments given in the proof of.i/ after equation (2.4) are still valid in the present situation and hence repeating the same process, we the required result..iii/if F D 0; then uv D 0; for all u;v 2 U and hence uœv;w D 0 for all u;v;w 2 U. Now using the similar arguments as used in the proof of.ii/ that follows equation (2.5), we get the required result. Therefore, we shall assume that d 0: For any u;v 2 U, we have Replacing v by 2vw; in (2.9) and using (2.9), we arrive at Œd.u/;F.v/ D uv: (2.9) F.v/Œd.u/;w C Œd.u/;v d.w/ C v Œd.u/;d.w/ D 0 for all u;v;w 2 U: For any u 2 L, replace w by d.u/; we obtain that Œd.u/;v d 2.u/ C v d.u/;d 2.u/ D 0 for all u 2 L;v 2 U: (2.10) Now, replace v by 2vw; in (2.10), to get 2 Œd.u/;v wd 2.u/ C v Œd.u/;w d 2.u/ C vw d.u/;d 2.u/ D 0; for all u 2 L;v;w 2 U. In the view of (2.10) the above expression yields that 2Œd.u/;v wd 2.u/ D 0; for all u 2 L;v;w 2 U: Since charr 2; we find that Œd.u/;v wd 2.u/ D 0 for all u 2 L;v;w 2 U: that is, Œd.u/;v Ud 2.u/ D 0 for all v 2 U;u 2 L; and hence using the similar arguments as used in the proof of.i/ that follows equation (2.4), we find the required result..iv/ If F D 0; then uv D 0; for all u;v 2 U: Using the same arguments as we used in the proof of.iii/, we get the required result. Therefore, we shall assume that d 0: For any u;v 2 U, we have.d.u/ ı F.v// D uv: (2.11) Replacing v by 2vw; in (2.11) and using (2.11), we obtain that F.v/Œd.u/;w C Œd.u/;v d.w/ C v.d.u/od.w// D 0 for all u;v;w 2 U: Now, replace w by d.u/ for all u 2 L, to get Œd.u/;v d 2.u/ C v d.u/od 2.u/ D 0 for all u 2 L;v 2 U: Now, applying similar technique as the one used after equation (2.10) in the proof of.iii/, we get the required result.

122 EMINE KOÇ AND NADEEM UR REHMAN Theorem 2. Let R be a prime ring with characteristic not two and U be a nonzero square closed Lie ideal of R: Suppose that R admits a generalized derivation F associated with nonzero derivation d which commutes with such that.i/ d.u/f.v/ D Œu;v for all u;v 2 U; or.ii/ d.u/f.v/ D u ı v for all u;v 2 U: If F D 0 or d 0, then U Z: Proof..i/If F D 0; then Œu;v D 0 for all u;v 2 U: Thus, by Lemma 3, we obtain that U Z: Suppose on contrary that U 6 Z. Therefore, we shall assume that d 0: We have Replacing u by 2uw;w 2 U in (2.12), we get d.u/f.v/ Œu;v D 0 for all u;v 2 U: (2.12) 2.d.u/wF.v/ C ud.w/f.v/ uœw;v Œu;v w/ D 0. Using (2.12) and charr 2; we see that d.u/wf.v/ Œu;v w D 0 for all u;v;w 2 U: (2.13) Substituting v for u in (2.13), we have That is, Using d D d, we get d.u/wf.u/ D 0 for all u;w 2 U: d.u/uf.u/ D 0 for all u 2 U:.d.u// UF.u/ D 0; for all u 2 U \ S a.r/: By Lemma 1, we get either d.u/ D 0 or F.u/ D 0; for each u 2 U \ S a.r/: Let u 2 U; as ucu ;u u 2 U \S a.r/ and d.uu / D 0 or F.uu / D 0: Hence, we obtain that d.u/ D 0 or F.u/ D 0; for all u 2 U: Let and K D fu 2 U j d.u/ D 0g L D fu 2 U j F.u/ D 0g of additive subgroups of U: Now using the same argument, we get U D K or U D L: If U D K; then U Z by Lemma 4, a contradiction. If U D L; then 0 D F.uv/ D F.u/v C ud.v/ D ud.v/; and so Ud.U / D.0/: Hence U Z by Lemma 5, a contradiction. This completes the proof..ii/ If F D 0; then uov D 0; for all u;v 2 U; and hence using the same arguments as used in the proof of Theorem 1.ii/, we get U Z.

NOTES ON GENERALIZED DERIVATIONS OF -PRIME RINGS 123 Suppose on the contrary that U 6 Z. Henceforth, we shall assume that d 6D 0. We have d.u/f.v/ u ı v D 0 for all u;v 2 U: (2.14) Writing 2uw;w 2 U by u in (2.14) and using charr 2, we find that Applying (2.14), we obtain d.u/wf.v/ C ud.w/f.v/ u.wov/ C Œu;v w D 0: d.u/wf.v/ C Œu;v w D 0 for all u;v;w 2 U: Using the same arguments as used in the proof of.i/ after equation (2.13), we get the required result. REFERENCES [1] M. Ashraf, A. Ali, and R. Rani, On generalized derivations of prime rings, Southeast Asian Bull. Math., vol. 29, no. 4, pp. 669 675, 2005. [2] M. Ashraf and A. Khan, Commutativity of -prime rings with generalized derivations, Rend. Semin. Mat. Univ. Padova, vol. 125, pp. 71 79, 2011. [3] Nadeem ur Rehman and R. M. Al-Omary, On commutativity of 2-torsion free *-prime rings with generalized derivations, Mathematica, vol. 53, no. 2, pp. 171 180, 2011. [4] L. Oukhtite and S. Salhi, On commutativity of -prime rings, Glas. Mat., III. Ser., vol. 41, no. 1, pp. 57 64, 2006. [5] L. Oukhtite and S. Salhi, Lie ideals and derivations of -prime rings, Int. J. Algebra, vol. 1, no. 1-4, pp. 25 30, 2007. [6] L. Oukhtite and S. Salhi, On generalized derivations of -prime ring, Afr. Diaspora J. Math., vol. 5, no. 1, pp. 21 25, 2007. [7] L. Oukhtite and S. Salhi, Centralizing automorphisms and Jordan left derivations on -prime rings, Adv. Algebra, vol. 1, no. 1, pp. 19 26, 2008. [8] L. Oukhtite, S. Salhi, and L. Taoufiq, Commutativity conditions on derivations and Lie ideals in -prime rings, Beitr. Algebra Geom., vol. 51, no. 1, pp. 275 282, 2010. Authors addresses Emine Koç Cumhuriyet University, Faculty of Science, Department of Mathematics, 58140, Sivas - TURKEY E-mail address: eminekoc@cumhuriyet.edu.tr Nadeem Ur Rehman Aligarh Muslim University,, Aligarh-India E-mail address: rehman100@gmail.com

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