Let R be a 2-torsion free σ-prime ring having a σ-square closed Lie ideal U and an automorphism T centralizing on U. We prove that if there exists u0 in Saσ(R) with Ru0 ⊂ U and if T commutes with σ on U, then U is contained in the center of R. This result is then applied to generalize the result of J. Mayne for centralizing automorphisms to σ-prime rings. Finally, for a 2-torsion free σ-prime ring possessing a nonzero derivation, we give suitable conditions under which the ring must be commutative

L. Oukhtite

S. Salhi

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GLASNIK MATEMATICKIVol. 41(61)(2006), 57 { 64ON COMMUTATIVITY OF -PRIME RINGSL. Oukhtite and S. SalhiUniversite Moulay Ismal and Universite Sidi Mohamed Ben Abdellah,MarocAbstract. Let R be a 2-torsion free -prime ring having a -squareclosed Lie ideal U and an automorphism T centralizing on U: We provethat if there exists u0 in Sa(R) with Ru0  U and if T commutes with on U; then U is contained in the center of R: This result is then ap-plied to generalize the result of J. Mayne for centralizing automorphismsto -prime rings. Finally, for a 2-torsion free -prime ring possessing anonzero derivation, we give suitable conditions under which the ring mustbe commutative.1. IntroductionA linear mapping T from a ring to itself is called centralizing on a subsetS of the ring if [x; T (x)] is in the center of the ring for every x in S: In partic-ular, if T satises [x; T (x)] = 0 for all x in S then T is called commuting onS: In [6] Posner showed that if a prime ring has a nontrivial derivation whichis centralizing on the entire ring, then the ring must be commutative. In [2]the same result is proved for a prime ring with a nontrivial centralizing auto-morphism. A number of authors have generalized these results by consideringmappings which are only assumed to be centralizing on an appropriate idealof the ring. In [1] Awtar considered centralizing derivations on Lie and Jordanideals. In the Jordan case, he proved that if a prime ring of characteristic nottwo has a nontrivial derivation which is centralizing on a Jordan ideal, thenthe ideal must be contained in the center of the ring. This result is extended in[3] where it is shown that if R is any prime ring with a nontrivial centralizingautomorphism or derivation on a nonzero ideal or (quadratic) Jordan ideal,2000 Mathematics Subject Classication. 16W10, 16W25, 16W20, 16U80.Key words and phrases. Rings with involution, -prime rings, centralizing automor-phisms, square closed Lie ideals, derivations, commutativity.5758 L. OUKHTITE AND S. SALHIthen R is commutative. For prime rings Mayne, in [4], also showed that a non-trivial automorphism which is centralizing on a Lie ideal implies that the idealis contained in the center if the ring is not of characteristic two. In this paper,the corresponding result for -prime rings with -square closed Lie ideals isproved, where  is an involution, Theorem 2.4. An immediate consequenceof Theorem 2.4 and the fact that a -ideal is a  square closed Lie ideal isTheorem 2.5 which extends the result of [3] for centralizing automorphisms to-prime rings of characteristic not two. To end this paper, for a 2-torsion free-prime ring having a nonzero derivation we give suitable conditions underwhich the ring must be commutative, Theorem 3.2 and Theorem 3.3.Throughout, R will represent an associative ring with center Z(R): Wesay R is 2-torsion free if for x 2 R; 2x = 0 implies x = 0: As usual thecommutator xy yx will be denoted by [x; y]: We shall use basic commutatoridentities [x; yz] = y[x; z] + [x; y]z; [xy; z] = x[y; z] + [x; z]y: An involution of a ring R is an anti-automorphism of order 2 (i.e.  is an additive mappingsatisfying (xy) = (y)(x) and 2(x) = x for all x; y 2 R): If R is equippedwith an involution ; we set Sa(R) := fr in R such that (r) = rg: Recallthat R is -prime if aRb = aR(b) = 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 pairs x; y 2 R: A Lie ideal U of R is called a square closed Lieideal if u2 2 U for all u 2 U and a -square closed Lie ideal if U is invariantunder : The fact that (u + v)2 2 U together with [u; v] 2 U yield 2uv 2 Ufor all u; v 2 U:2. Automorphisms centralizing on -square closed Lie idealsThroughout this section R will denote a 2-torsion free -prime ring, where is an involution of R:Lemma 2.1. If T is an homomorphism of R which is centralizing on asquare closed Lie ideal U; then T is commuting on U:Proof. By linearization [x; T (y)] + [y; T (x)] is in Z(R) for all x and y inU: Thus, [x; T (x2)] + [x2; T (x)] in Z(R) and thereforeT (x)[x; T (x)] + [x; T (x)]x + x[x; T (x)] + [x; T (x)]x = 2(x+ T (x))[x; T (x)]in Z(R): Since R is 2-torsion free, then(x + T (x))[x; T (x)] in Z(R):For r in R; we then getr(x + T (x))[x; T (x)] = (x+ T (x))[x; T (x)]r = (x+ T (x))r[x; T (x)]:Hence [r; x+ T (x)][x; T (x)] = 0 for all r in R: In particular,0 = [x; x+ T (x)][x; T (x)] = [x; T (x)]2:ON COMMUTATIVITY OF -PRIME RINGS 59Since [x; T (x)] in Z(R); then[x; T (x)]R[x; T (x)] = 0:Therefore,[x; T (x)]R[x; T (x)]([x; T (x)]) = 0and since [x; T (x)]([x; T (x)]) is invariant under ; the -primeness of R yields[x; T (x)] = 0 or [x; T (x)]([x; T (x)]) = 0: If [x; T (x)]([x; T (x)]) = 0 then[x; T (x)]R([x; T (x)]) = 0; because [x; T (x)] 2 Z(R)and consequently[x; T (x)]R[x; T (x)] = [x; T (x)]R([x; T (x)]) = 0:Once again using the -primeness of R; we then get [x; T (x)] = 0 for all x inU; hence T is commuting on U:From now on assume that T is an automorphism centralizing on a -square closed Lie ideal U which contains an element u0 in Sa(R) such thatRu0  U: Since T is centralizing on U; Lemma 2.1 implies [x; T (x)] = 0 forall x in U:Lemma 2.2. If a; b in R are such that aUb = aU(b) = 0; then a = 0 orb = 0:Proof. Suppose a 6= 0: We have to distinguish two cases:1) u0 in Z(R): Let r in R. From aru0b = aru0(b) = 0 it follows thataRu0b = aRu0(b) = aR(u0b) = 0so that u0b = 0: Since u0 is central, then u0Rb = (u0)Rb = 0 proving b = 0:2) u0 62 Z(R): If a[t; u0] = 0 for all t in R; thena[tr; u0] = at[r; u0] = 0 so that aR[r; u0] = 0 = aR([r; u0])proving [r; u0] = 0 for all r in R which contradicts u0 62 Z(R): Thus thereexists t in R such that a[t; u0] 6= 0: Froma[t; u0]rb = a[t; u0]r(b) = 0;it follows thata[t; u0]Rb = a[t; u0]R(b) = 0and the -primeness of R yields b = 0:Lemma 2.3. Suppose that T commutes with  on U: If x in U \ Sa(R)satises T (x) 6= x, then x in Z(R):60 L. OUKHTITE AND S. SALHIProof. Let x in U \ Sa(R) with T (x) 6= x: From [t; T (t)] = 0; forall t in U; we conclude [t; T (y)] = [T (t); y] for all t; y in U: In particular[x; T (2xy)] = [T (x); 2xy]; because 2xy in U: Since R is 2-torsion free, thusT (x)[x; T (y)]  x[T (x); y] = 0and therefore(T (x)  x)[T (x); y] = 0 for all y in U:For u in U; as 2uy in U and once again using the fact that R is 2-torsion freewe obtain0 = (x  T (x))[T (x); uy] = (x  T (x))u[T (x); y]:Hence(x  T (x))U [T (x); y] = (x  T (x))U([T (x); y]) = 0:Applying Lemma 2.2, since T (x) 6= x; then [T (x); y] = 0 for all y in U: Whence[T (x); tru0] = [T (x); t]ru0 = 0 for all r; t in R:Thus [T (x); t]Ru0 = 0; which proves [T (x); t] = 0 so that T (x) in Z(R): SinceT is an automorphism then x in Z(R):Theorem 2.4. Let R be a 2-torsion free -prime ring having an automor-phism T 6= 1 centralizing on a  square closed Lie ideal U: If T commuteswith  on U and there exists u0 in Sa(R) with Ru0  U; then U is containedin Z(R):Proof. Suppose that T is identity on U; hence for all t; r in R we thengetT (tru0) = tru0 = T (t)T (ru0) = T (t)ru0:Thus(T (t)  t)ru0 = 0 so that (T (t)  t)Ru0 = 0:Since R is -prime this yields T (t) = t for all t in R which is impossible.Thus T is nontrivial on U: Since R is 2-torsion free, the fact that x + (x)and x  (x) are in U \ Sa(R) for all x in U assures that T is nontrivial onU \ Sa(R): Therefore, there must be an element x in U \ Sa(R) such thatx 6= T (x) and x is then in Z(R) by Lemma 2.3. Let 0 6= y be in U \ Sa(R)and not be in Z(R): Once again using Lemma 2.3, we obtain T (y) = y: ButthenT (xy) = T (x)y = xy so that (T (x)  x)y = 0and therefore(T (x)  x)Ry = (T (x)  x)R(y) = 0; because x in Z(R):As R is -prime this yields y = 0: Hence for all y in U \ Sa(R), y must bein Z(R): Now let x in U . The fact that x   (x) and x + (x) are elementsin U \ Sa(R) gives x   (x) and x + (x) in Z(R) and thus 2x in Z(R):Consequently, x in Z(R) which proves U  Z(R):ON COMMUTATIVITY OF -PRIME RINGS 61In [3] it is proved that if a prime ring has a nontrivial automorphismwhich centralizes on a nonzero ideal, then the ring is commutative. Thepurpose of the following theorem is to generalize this result to -prime ringswith characteristic not two.Theorem 2.5. Let R be a 2-torsion free -prime ring having an auto-morphism T 6= 1 which commutes with  on a nonzero -ideal J of R: If T iscentralizing on J; then R is a commutative ring.Proof. Since a -ideal is a -square closed Lie ideal, from Theorem 2.4it follows that J is contained in Z(R): Now, if x2 = 0 for all x 2 J , then((x) + x)2 = 0: As (x) + x is invariant under ; the fact that ((x) +x)R((x) + x) = 0 together with the -primeness of R yield (x) =  x: Butx2 = 0 implies xRx = 0 so that x = 0 which contradicts J 6= 0: Thus thereexists an element x 2 J such that x2 6= 0: For all r; s 2 R; we havex2rs = x(xr)s = xrxs = x(rx)s = rxxs = xsrx = x2sr:Hence x2(rs   sr) = 0 so that x2R[r; s] = 0 and similarly x2R([r; s]) = 0:Since x2 6= 0, the -primeness of R gives [r; s] = 0 for all r; s 2 R; proving thecommutativity of R:3. Derivations in -prime ringsLet R be a 2-torsion free -prime ring and let d be a nonzero derivationon R: Our aim in this section is to give suitable conditions under which thering R must be commutative. We will make frequent and important uses ofthe following lemma.Lemma 3.1 ([5], 3) of Theorem 1). Let I be a nonzero -ideal of R: Ifa; b in R are such that aIb = 0 = aI(b); then a = 0 or b = 0:Proof. Suppose a 6= 0; there exists some x 2 I such that ax 6= 0: Indeed,otherwiseaRx = 0 and aR(x) = 0 for all x 2 Iand therefore a = 0: Since aIRb = 0 and aIR(b) = 0; we then obtainaxRb = axR(b) = 0:In view of the -primeness of R this yields b = 0:Theorem 3.2. Let 0 6= d be a derivation of R and let I be a nonzero-ideal of R: If r in Sa(R) satises [d(x); r] = 0 for all x in I, then r inZ(R): Furthermore, if d(I)  Z(R); then R is commutative.Proof. Since [d(uv); r] = 0 for all u; v in I; it follows thatd(u)vr + ud(v)r   rd(u)v   rud(v) = 0:62 L. OUKHTITE AND S. SALHIUsing [d(u); r] = [d(v); r] = 0; we obtain(3.1) d(u)[v; r] + [u; r]d(v) = 0 for all u; v 2 I:Replacing v by vr in (3.1), we conclude that [u; r]Id(r) = 0: The fact that Iis a -ideal together with r in Sa(R); give([u; r])Id(r) = [u; r]Id(r) = 0:Applying Lemma 3.1, either d(r) = 0 or [u; r] = 0: If d(r) 6= 0; then [u; r] = 0for all u in I: Let t in R; from [tu; r] = 0 it follows that [t; r]u = 0: Let 0 6= x0in I; as[t; r]Rx0 = [t; r]R(x0) = 0then [t; r] = 0; since R is -prime, which proves r in Z(R):Now if d(r) = 0; then d([u; r]) = [d(u); r] = 0 and consequently(3.2) d([I; r]) = 0:Replace v by v! in (3.1), where ! in I; we have(3.3) d(u)v[!; r] + [u; r]vd(!) = 0:Taking [!; r] instead of ! in (3.3) and applying (3.2) we then getd(u)v[[!; r]; r] = 0 so that d(u)I [[!; r]; r] = 0 = d(u)I([[!; r]; r])whence d(I) = 0 or [[!; r]; r] = 0 for all ! in I; by Lemma 3.1.If d(I) = 0; then for any t in R we get d(tu) = d(t)u = 0 for all u in I:Therefored(t)RI = d(t)R(I) = 0and as 0 6= I; then d(t) = 0 in such a way that d = 0: Consequently,(3.4) [[!; r]; r] = 0:Replace ! by !u in (3.4) we obtain0 = [[!u; r]; r] = [!; r][u; r] + [!; r][u; r]in such a way that [!; r][u; r] = 0; because R is 2-torsion free. Hence0 = [t!; r][u; r] = [t; r]![u; r]and consequently[t; r]I [u; r] = 0 for all u in I:Therefore[t; r]I [u; r] = [t; r]I([u; r]) = 0;once again using Lemma 3.1, we see that [t; r] = 0 or [u; r] = 0: If [t; r] = 0;then r in Z(R): If [u; r] = 0 for all u in I; then for any t 2 R0 = [tu; r] = t[u; r] + [t; r]u = [t; r]u:Hence0 = [t; r]I = [t; r]I1 = [t; r]I(1):ON COMMUTATIVITY OF -PRIME RINGS 63Using Lemma 3.1 we conclude that [t; r] = 0; which proves that r in Z(R):Now suppose that d(I)  Z(R) and let r in R: From the rst part of thetheorem we conclude Sa(R)  Z(R): Using the fact thatr + (r) and r   (r) are elements of Sa(R)we then obtainr   (r) 2 Z(R) and r + (r) 2 Z(R)and hence 2r in Z(R): Since R is 2-torsion free, then r in Z(R) proving thecommutativity of R:Theorem 3.3. Let d be a nonzero derivation of R and let a in Sa(R):If d([R; a]) = 0; then a in Z(R): In particular, if d(xy)   d(yx) = 0; for allx; y 2 R; then R is a commutative ring.Proof. If d(a) = 0; from our hypothesis, we have for any r in R;0 = d([r; a]) = d(r)a + rd(a)   d(a)r   ad(r) = d(r)a   ad(r) = [d(r); a]:Therefore[d(r); a] = 0 for all r in R:Applying Theorem 3.2, this yields a in Z(R) and the proof is then complete.Now, assume that d(a) 6= 0: For all r in R;0 = d([ar; a]) = d(a[r; a]) = d(a)[r; a] + ad([r; a])and so,(3.5) d(a)[r; a] = 0:Taking rs; s in R instead of r in (3.5), we obtain0 = d(a)[rs; a] = d(a)r[s; a] + d(a)[r; a]s:Using (3:5); this yields d(a)r[s; a] = 0 so thatd(a)R[s; a] = 0 for all s in R:Since a in Sa(R); then0 = d(a)R[s; a] = d(a)R([s; a])and the -primeness of R yields [s; a] = 0 which proves a in Z(R):Now, assume that d([x; y]) = 0 for all x; y 2 R: Applying the rst part ofour theorem, we then get Sa(R)  Z(R): For r in R; the fact thatr + (r) and r   (r) are elements of Sa(R);yields 2r in Z(R): Since R is 2-torsion free, this yields r in Z(R) which provesthat R is a commutative ring.64 L. OUKHTITE AND S. SALHIReferences[1] R. Awtar, Lie and Jordan structure in prime rings with derivations, Proc. Amer. Math.Soc. 41 (1973), 67-74.[2] J. Mayne, Centralizing automorphisms of prime rings, Canad. Math. Bull. 19 (1976),113-115.[3] J. Mayne, Centralizing mappings of prime rings, Canad. Math. Bull. 27 (1984), 122-126.[4] J. Mayne, Centralizing automorphisms of Lie ideals in prime rings, Canad. Math. Bull.35 (1992), 510-514.[5] L. Oukhtite and S. Salhi, -prime rings with a special kind of automorphism, submitted.[6] E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093-1100.L. OukhtiteUniversite Moulay Ismal, Faculte des Sciences et TechniquesDepartement de MathematiquesB. P. 509-Boutalamine, ErrachidiaMarocE-mail : oukhtite@math.netS. SalhiUniversite Sidi Mohamed Ben Abdellah, Faculte des SciencesDepartement de Mathematiques et InformatiqueB. P. 1796-Atlas, FesMarocE-mail : salhi@math.netReceived : 20.7.2005.Revised : 22.10.2005.

On commutativity of σ-prime rings

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