The concept of derivations as well as of generalized inner derivations have been generalized as an additive function F : R &#8594; R satisfying F(xy) = F(x)y + xd(y) for all x, y &#8712; R, where d is a derivation on R, such a function F is said to be a generalized derivation. In the present paper we have discussed the commutativity of prime rings admitting a generalized derivation F satisfying (i) [F(x), x] = 0, (ii) F([x, y]) = [x, y], and (iii) F(x &#9702; y) = x &#9702; y for all x, y in some appropriate subset of R.</p

Rehman, Nadeem ur

Okayama University Scientific Achievement Repository

Mathematical Journal of OkayamaUniversityVolume 44, Issue 1 2002 Article 4JANUARY 2002On Commutativity of Rings with GeneralizedDerivationsNadeem ur Rehman∗∗University KaiserslauternCopyright c©2002 by the authors. Mathematical Journal of Okayama University is produced byThe Berkeley Electronic Press (bepress). http://escholarship.lib.okayama-u.ac.jp/mjouOn Commutativity of Rings with GeneralizedDerivationsNadeem ur RehmanAbstractThe concept of derivations as well as of generalized inner derivations have been generalizedas an additive function F : R→ R satisfying F(xy) = F(x)y + xd(y) for all x, y ∈ R, where d is aderivation on R, such a function F is said to be a generalized derivation. In the present paper wehave discued the commutativity of prime rings admitting a generalized derivation F satisfying (i)[F(x), x] = 0, (ii) F([x, y]) = [x, y], and (iii) F(x &#x25E6; y) = x &#x25E6; y for all x, y in someappropriate subset of R.KEYWORDS: prime rings, generalized derivations, derivations, ideals, Lie ideals and commuta-tivity.Math. J. Okayama Univ. 44(2002), 43{49ON COMMUTATIVITY OF RINGSWITH GENERALIZED DERIVATIONSNadeem-ur-REHMANAbstract. The concept of derivations as well as of generalized innerderivations have been generalized as an additive function F : R ! Rsatisfying F (xy) = F (x)y + xd(y) for all x; y 2 R, where d is a deriva-tion on R, such a function F is said to be a generalized derivation. Inthe present paper we have discussed the commutativity of prime ringsadmitting a generalized derivation F satisfying (i) [F (x); x] = 0, (ii)F ([x; y]) = [x; y], and (iii) F (x ± y) = x ± y for all x; y in some appropri-ate subset of R.1. IntroductionLet R denote an associative ring with center Z(R). For any x; y 2 R,the symbol [x; y] stands for the commutator xy ¡ yx and the symbol x ± ydenotes for anti-commutator xy + yx. Recall that a ring R is called primeif for any a; b 2 R, aRb = (0) implies that either a = 0 or b = 0. Anadditive mapping d : R ! R is called a derivation if d(xy) = d(x)y + xd(y)holds for all x; y 2 R. For a ¯xed a 2 R, the mapping Ia : R ! R given byIa(x) = [a; x] is a derivation which is said to be inner derivation.Many analysts have studied generalized derivation in the context of al-gebras on certain normed spaces (see [10] for reference). By a generalizedderivation on an algebra A one usually means a map of the form x 7! ax+xb,where a and b are ¯xed elements in A. We prefer to call such maps gen-eralized inner derivations for the reason they present a generalization ofthe concept of inner derivations (i.e. the maps of the form x 7! ax ¡ xa).Now in a ring R, let F be a generalized inner derivation of R given byF (x) = ax+xb. Notice that F (xy) = F (x)y+xIb(y), where Ib(y) = yb¡ byis an inner derivation.Motivated by these observation Hvala [10] introduced the notions of gen-eralized derivations in rings. An additive mapping F : R ! R is calleda generalized derivation if there exists a derivation d : R ! R such thatF (xy) = F (x)y + xd(y) holds for all x; y 2 R. Obviously, every derivationgeneralized inner derivation and left multiplier (i.e. an additive mappingF : R! R such that F (xy) = F (x)y for all x; y 2 R) are generalized deriva-tions.1991 Mathematics Subject Classi¯cation. 16W25, 16N60, 16U80.Key words and phrases. prime rings, generalized derivations, derivations, ideals, Lieideals and commutativity.431Rehman: On Commutativity of Rings with Generalized DerivationsProduced by The Berkeley Electronic Press, 200244 N. REHMANIn the present paper we shall attempt to generalize some known resultsfor derivations to generalized derivations.2. Preliminary resultsThroughout the present paper, we shall make use of the following twobasic identities without any speci¯c mention:x ± (yz) = (x ± y)z ¡ y[x; z] = y(x ± z) + [x; y]z;(xy) ± z = x(y ± z)¡ [x; z]y = (x ± z)y + x[y; z]:We begin with the following known results which will be used extensivelyto prove our theorems.Lemma 2.1 ([4, Lemma 4]). If U 6µ Z(R) is Lie ideal of a 2-torsion freeprime ring R and a; b 2 R such that aUb = 0, then a = 0 or b = 0.Lemma 2.2 ([4, Lemma 5]). Let R be a 2-torsion free prime ring and U anonzero Lie ideal of R. If d is a nonzero derivation of R such that d(U) = 0,then U µ Z(R).Lemma 2.3 ([2, Theorem 7]). Let R be a 2-torsion free prime ring andU a nonzero Lie ideal of R. If d is a nonzero derivation of R such that[u; d(u)] 2 Z(R) for all u 2 U , then U µ Z(R).Lemma 2.4 ([3, Theorem 4]). Let R be a prime ring and I a nonzero leftideal of R. If R admits a nonzero derivation d such that [x; d(x)] is centralfor all x 2 I, then R is commutative.Lemma 2.5 ([11, Lemma 3]). If a prime ring R contains a nonzero com-mutative right ideal, then R is commutative.Now, we prove the following.Lemma 2.6. Let R be a 2-torsion free prime ring and U be a nonzero Lieideal of R. If U is a commutative Lie ideal of R, i.e. [u; v] = 0 for allu; v 2 U , then U µ Z(R).Proof. Since U is a commutative Lie ideal of R, i.e.(2.1) [u; v] = 0; for all u; v 2 U:Replacing v by [u; r] in (2.1), we get [u; [u; r]] = 0 for all u 2 U , r 2 R.Again replace r by rs, to get [u; [u; rs]] = 0 for all u 2 U , r; s 2 R, that is[u; [u; r]]s+ r[u; [u; s]] + 2[u; r][u; s] = 0; for all u 2 U; r; s 2 R:This implies that 2[u; r][u; s] = 0 for all u 2 U , r; s 2 R. Since char(R) 6= 2,we get [u; r][u; s] = 0. Replacing s by sr, we get [u; r]R[u; r] = (0) for allu 2 U , r 2 R. Thus primeness of R forces that [u; r] = 0 for all u 2 U ,r 2 R, and hence U µ Z(R). ¤2Mathematical Journal of Okayama University, Vol. 44 [2002], Iss. 1, Art. 4http://escholarship.lib.okayama-u.ac.jp/mjou/vol44/iss1/4ON COMMUTATIVITY OF RINGS WITH GENERALIZED DERIVATIONS 453. Lie ideals and generalized derivations of prime ringsTheorem 3.1. Let R be a 2-torsion free prime ring and U be a nonzero Lieideal of R such that u2 2 U for all u 2 U . If R admits a nonzero generalizedderivation F with d such that [F (u); u] = 0 for all u 2 U , and if d 6= 0, thenU µ Z(R).Proof. We have(3.1) [F (u); u] = 0; for all u 2 U:Linearizing (3.1) and using (3.1), we obtain(3.2) [F (u); v] + [F (v); u] = 0; for all u; v 2 U:Notice that vw+wv = (v+w)2¡ v2¡w2 for all v; w 2 U . Since u2 2 U forall u 2 U , vw + wv 2 U . Also vw ¡ wv 2 U for all v; w 2 U . Hence we ¯ndthat 2vw 2 U for all v; w 2 U . Replacing v by 2vu in (3.2) and use (3.1)and (3.2), to get(3.3) v[d(u); u] + [v; u]d(u) = 0; for all u; v 2 U:Again replacing v by 2wv in (3.3) and using (3.3), we get [w; u]vd(u) = 0for all u; v; w 2 U , and hence [w; u]Ud(u) = (0) for all u;w 2 U . Thus foreach u 2 U , by Lemma 2.1 we ¯nd that either [w; u] = 0 or d(u) = 0. Now,let A = fu 2 U j d(u) = 0g and B = fu 2 U j [w; u] = 0 for all w 2 Ug.Then A and B are additive subgroups of U and U = A [ B. But a groupcan not be a union of two its proper subgroups, and hence U = A or U = B.If U = A, then d(u) = 0 for all u 2 U . Thus by Lemma 2.2, we get therequired result. On the other hand if [w; u] = 0 for all w; u 2 U , then byLemma 2.6, we get U µ Z(R). This completes the proof of the theorem. ¤Using the same techniques with necessary variations, we can prove thefollowing corollary even without the characteristic assumption on the ring.Corollary 3.2. Let R be a prime ring. If R admits a nonzero generalizedderivation F with d such that [F (x); x] = 0 for all x 2 R, and if d 6= 0, thenR is commutative.In a recent paper, Daif and Bell [7] established that a semiprime ring Rmust be commutative if it admits a derivation d such that d([x; y]) = [x; y]for all x; y 2 R. Further, Ashraf and Rehman [1] extended the mentionedresult for Lie ideals of R. In the present section we generalize this result forgeneralized derivations and Lie ideals of R.Theorem 3.3. Let R be a 2-torsion free prime ring and U a nonzero Lieideal of R such that u2 2 U for all u 2 U . If R admits a generalizedderivation F with d such that F ([u; v]) = [u; v] for all u; v 2 U , and if F = 0or d 6= 0, then U µ Z(R).3Rehman: On Commutativity of Rings with Generalized DerivationsProduced by The Berkeley Electronic Press, 200246 N. REHMANProof. Given that F is a generalized derivation of R such that F ([u; v]) =[u; v] for all u; v 2 U . If F = 0, then [u; v] = 0 for all u; v 2 U . Thus byLemma 2.6, we get the required result.Now, onward we assume that F 6= 0. Suppose on contrary that U 6µ Z(R).For any u; v 2 U , we have F ([u; v]) = [u; v]. This can be rewritten as(3.4) F (u)v + ud(v)¡ F (v)u¡ vd(u) = [u; v]; for all u; v 2 U:Replacing v by 2vu in (3.4) and using the fact that char(R) 6= 2, we ¯ndthatF (u)vu+ ud(v)u+ [u; v]d(u)¡F (v)u2¡ vd(u)u = [u; v]u; for all u; v 2 U;and hence application of (3.4) gives that [u; v]d(u) = 0 for all u; v 2 U .Again replace v by 2wv, to get [u;w]vd(u) = 0 for all u; v; w 2 U , and hence[u;w]Ud(u) = (0) for all u;w 2 U . Thus for each u 2 U , by Lemma 2.1,either [u;w] = 0 or d(u) = 0. Now, let U1 = fu 2 U j [u;w] = 0 for all w 2Ug and U2 = fu 2 U j d(u) = 0g. Then U1 and U2 both are additivesubgroups of U and U1 [ U2 = U . Thus either U1 = U or U2 = U . IfU1 = U , then [u;w] = 0 for all u;w 2 U . Hence by Lemma 2.6, we getU µ Z(R), contradiction. On the other hand if U2 = U , then d(u) = 0 forall u 2 U . Thus by Lemma 2.2, we get U µ Z(R), again a contradiction.This completes the proof of the theorem. ¤Using the same techniques with necessary variations we get the following.Theorem 3.4. Let R be a 2-torsion free prime ring and U a nonzero Lieideal of R such that u2 2 U for all u 2 U . If R admits a generalizedderivation F with d such that F ([u; v]) + [u; v] = 0 for all u; v 2 U , and ifF = 0 or d 6= 0, then U µ Z(R).Corollary 3.5. Let R be a 2-torsion free prime ring and U a nonzero Lieideal of R such that u2 2 U for all u 2 U . If R admits a generalizedderivation F with d such that F (uv) = uv for all u; v 2 U , and if F = 0 ord 6= 0, then U µ Z(R).Proof. For any u; v 2 U , F (uv¡ vu) = F (uv)¡F (vu) = uv¡ vu, and henceby Theorem 3.3, we get the required result. ¤Similarly, in view of the Theorem 3.4, we get the following.Corollary 3.6. Let R be a 2-torsion free prime ring and U a nonzero Lieideal of R such that u2 2 U for all u 2 U . If R admits a generalizedderivation F with d such that F (uv) = vu for all u; v 2 U , and if F = 0 ord 6= 0, then U µ Z(R).4Mathematical Journal of Okayama University, Vol. 44 [2002], Iss. 1, Art. 4http://escholarship.lib.okayama-u.ac.jp/mjou/vol44/iss1/4ON COMMUTATIVITY OF RINGS WITH GENERALIZED DERIVATIONS 47Theorem 3.7. Let R be a 2-torsion free prime ring and U a nonzero Lieideal of R such that u2 2 U for all u 2 U . If R admits a generalizedderivation F with d such that F (u ± v) = u ± v for all u; v 2 U , and if F = 0or d 6= 0, then U µ Z(R).Proof. If F = 0, then we have(3.5) u ± v = 0; for all u; v 2 U:Replacing v by 2vw in (3.5) and using (3.5), we have 2v[u;w] = 0 for allu; v; w 2 U . This implies that v[u;w] = 0 for all u; v; w 2 U . Again replacev by [u; r], to get [u; r][u;w] = 0 for all u;w 2 U , r 2 R. For any s 2 R,replacing r by rs, we get [u; r]R[u;w] = (0) for all u;w 2 U , r 2 R. Thus,in particular we have [u;w]R[u;w] = (0) for all u;w 2 U . Thus primenessof R yields that [u;w] = 0 for all u;w 2 U , and hence by Lemma 2.6, we getthe required result.Therefore now onward we shall assume that F 6= 0. Suppose on contrarythat U 6µ Z(R). For any u; v 2 U , we have F (u ± v) = u ± v. This can berewritten as(3.6) F (u)v + ud(v) + F (v)u+ vd(u) = u ± v; for all u; v 2 U:Replacing v by 2vu in (3.6), we ¯nd that(F (u)v+ud(v)+F (v)u+vd(u)¡u±v)u+(u±v)d(u) = 0; for all u; v 2 U:Thus an application of (3.6) gives that (u ± v)d(u) = 0 for all u; v 2 U .Again replace v by 2wv, to get [u;w]vd(u) = 0 for all u; v; w 2 U . Notethat the arguments given in the proof of Theorem 3.3 are still valid in thepresent situation and hence repeating the same process we get the requiredresult. ¤Using similar arguments one can also prove the following.Theorem 3.8. Let R be a 2-torsion free prime ring and U a nonzero Lieideal of R such that u2 2 U for all u 2 U . If R admits a generalizedderivation F with d such that F (u ± v) + u ± v = 0 for all u; v 2 U , and ifF = 0 or d 6= 0, then U µ Z(R).4. Ideals and generalized derivations of prime ringsIn the hypothesis of Theorems 3.7 and 3.8, if we choose the underlyingsubset as an ideal instead of a Lie ideal, then we can prove the followingresult even without the characteristic assumption on the ring.Theorem 4.1. Let R be a prime ring and I a nonzero ideal of R. If Radmits a generalized derivation F with d such that F (x±y) = x±y holds forall x; y 2 I, and if F = 0 or d 6= 0, then R is commutative.5Rehman: On Commutativity of Rings with Generalized DerivationsProduced by The Berkeley Electronic Press, 200248 N. REHMANProof. For any x; y 2 I, we have F (x±y) = x±y. If F = 0, then x±y = 0 forall x; y 2 I. Replacing y by yz and using the fact that xy = ¡yx, we ¯ndthat y[x; z] = 0 for all x; y; z 2 I, and hence IR[x; z] = (0) for all x; z 2 I.Since I 6= (0) and R is prime, we get [x; z] = 0 for all x; z 2 I, and hence byLemma 2.5, R is commutative. Hence onward we assume that F 6= 0. Forany x; y 2 I, we have F (x ± y) = x ± y. This can be rewritten as(4.1) F (x)y + xd(y) + F (y)x+ yd(x) = x ± y; for all x; y 2 I:Replacing y by yx in (4.1), we get(F (x)y+xd(y)+F (y)x+yd(x)¡(x±y))x+(x±y)d(x) = 0; for all x; y 2 I:In view of (4.1) the above relation yields that (x±y)d(x) = 0 for all x; y 2 I.Again replace y by zy, to get z(x±y)d(x)+ [x; z]yd(x) = 0 for all x; y; z 2 I,and hence [x; z]IRd(x) = (0) for all x; z 2 I. Thus primeness of R forcesthat for each x 2 I either d(x) = 0 or [x; z]I = (0) for all z 2 I. The setof x 2 I for which these two properties hold are additive subgroups of Iwhose union is I, and therefore d(x) = 0 for all x 2 I or [x; z]I = (0) for allx; z 2 I. If [x; z]I = (0) for all x; z 2 I, then [x; z]RI = (0). Since I 6= (0),we ¯nd that [x; z] = 0 for all x; z 2 I, and hence again by Lemma 2.5, R iscommutative. On the other hand if d(x) = 0 for all x 2 I, then implies that[d(x); x] = 0 for all x 2 I, and hence by Lemma 2.4, R is commutative. ¤Using similar arguments as used in the above theorem, we can prove thefollowing.Theorem 4.2. Let R be a prime ring and I a nonzero ideal of R. If Radmits a generalized derivation F with d such that F (x±y)+x±y = 0 holdsfor all x; y 2 I, and if F = 0 or d 6= 0, then R is commutative.Remark. In view of the above results, it is an obvious question is whetherthese results can be extended to left multiplier (i.e. a generalized derivationwith d = 0). Unfortunately, we are unable to extend these results to thecase where F is a left multiplier. We leave as an open question whether ornot these results can be extended in the setting of left multiplier.AcknowledgmentThe author is greatly indebted to the referee for serval useful suggestionsand valuable comments.References[1] M. Ashraf and N. Rehman, On commutativity of rings with derivations, ResultsMath. (to appear).[2] R. Awtar, Lie structure in prime rings with derivations, Publ. Math. Debrecen31(1984), 209{215.6Mathematical Journal of Okayama University, Vol. 44 [2002], Iss. 1, Art. 4http://escholarship.lib.okayama-u.ac.jp/mjou/vol44/iss1/4ON COMMUTATIVITY OF RINGS WITH GENERALIZED DERIVATIONS 49[3] H. E. Bell and W. S. Martindale, Centralizing mappings of semiprime rings,Canad. Math. Bull. 30(1987), 92{101.[4] J. Bergen, I. N. Herstein and J. W. Kerr, Lie ideals and derivations of primerings, J. Algebra 71(1981), 259{267.[5] M. Bre·sar, Commuting traces of biadditive mappings, commutativity-preservingmappings and Lie mappings, Trans. Amer. Math. Soc. 335(1993), 525{546.[6] M. Bre·sar, Centralizing mappings and derivations in prime rings, J. Algebra156(1993), 385{394.[7] M. N. Daif and H. E. Bell, Remarks on derivations on semiprime rings, Internat.J. Math. Math. Sci. 15(1992), 205{206.[8] I. N. Herstein, Ring with involution, Univ. Chicago Press, Chicago, 1976.[9] I. N. Herstein, Topics in ring theory, Univ. Chicago Press, Chicago, 1969.[10] B. Hvala, Generalized derivations in rings, Comm. Algebra 26(1998), 1147{1166.[11] J. H. Mayne, Centralizing mappings of prime rings, Canad. Math. Bull. 27(1984),122{126.[12] E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8(1957), 1093{1100.Nadeem-ur-RehmanDepartment of MathematicsUniversity KaiserslauternP.O.Box. 304967653 Kaiserslautern, Germanye-mail address: rehman100@postmark.net(Received January 8, 2002 )7Rehman: On Commutativity of Rings with Generalized DerivationsProduced by The Berkeley Electronic Press, 2002

On Commutativity of Rings with Generalized Derivations

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