The aim of this paper is to define generalized ▫$(m, n)$▫-Jordan centralizers and to prove that on a prime ring with nonzero center and ▫${rm char}(R) ne 6mn(m+n)(m+2n)$▫ every generalized ▫$(m, n)$▫-Jordan centralizer is a two-sided centralizer.V članku je pokazano, da je vsak posplošeni ▫$(m,n)$▫-jordanski centralizator na prakolobarju z neničelnim centrom in karakteristiko različno od ▫$6mn(m+n)(m+2n)$▫ dvostranski centralizator

Fošner, Ajda

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DEMONSTRATIO MATHEMATICAVol. XLVI No 2 2013Ajda FošnerA NOTE ON GENERALIZED (m,n)-JORDANCENTRALIZERSAbstract. The aim of this paper is to define generalized (m,n)-Jordan centralizersand to prove that on a prime ring with nonzero center and char(R) 6= 6mn(m+n)(m+2n)every generalized (m,n)-Jordan centralizer is a two-sided centralizer.Throughout, R will represent an associative ring with a center Z(R).Let n ≥ 2 be an integer. A ring R is said to be n-torsion free if for x ∈ R,nx = 0 implies x = 0. Recall that R is prime if aRb = {0} implies a = 0 orb = 0. A ring R is called semiprime if aRa = {0} implies a = 0. An additivemapping D : R→ R is called a derivation if D(xy) = D(x)y + xD(y) holdsfor all x, y ∈ R and is called a Jordan derivation if D(x2) = D(x)x+ xD(x)is fulfilled for all x ∈ R. One can easily prove that every derivation is aJordan derivation, but converse is in general not true. A classical result dueto Herstein [5, Theorem 3.3] asserts that a Jordan derivation on a primering of characteristic different from two is a derivation. A brief proof ofHerstein’s result can be found in [3]. This result was extended to 2-torsionfree semiprime rings by Cusack [4] (see [2] for an alternative proof).An additive mapping T : R → R is called a left (right) centralizer ifT (xy) = T (x)y (T (xy) = xT (y)) holds for all pairs x, y ∈ R. If R has theidentity element, then T : R → R is a left centralizer if and only if T is ofthe form T (x) = ax for all x ∈ R, where a ∈ R is a fixed element. For asemiprime ring R, left centralizers are of the form T (x) = qx for all x ∈ R,where q is a fixed element of a Martindale right ring of quotients Qr (see,for example, Chapter 2 in [1]). An additive mapping T : R → R is called aleft (right) Jordan centralizer if T (x2) = T (x)x (T (x2) = xT (x)) holds forall x ∈ R. We call an additive mapping T : R → R a two-sided centralizer2010 Mathematics Subject Classification: 16N60, 39B05.Key words and phrases: prime ring, semiprime ring, left (right) centralizer, left (right)Jordan centralizer, (m,n)-Jordan centralizer, generalized (m,n)-Jordan centralizer.258 A. Fošner(a two-sided Jordan centralizer) if T is both a left and a right centralizer(a left and a right Jordan centralizer). If R is a semiprime ring with anextended centroid C and T : R → R is a two-sided centralizer, then thereexists an element λ ∈ C such that T (x) = λx for all x ∈ R (see Theorem2.3.2 in [1]). In [9], Zalar has proved that any left (right) Jordan centralizeron a 2-torsion free semiprime ring is a left (right) centralizer.In [8], Vukman defined (m,n)-Jordan centralizers in the following way.Definition 1. Let m ≥ 0, n ≥ 0 with m + n 6= 0 be some fixed integersand R an arbitrary ring. An additive mapping T : R → R is called a(m,n)-Jordan centralizer if(m+ n)T (x2) = mT (x)x+ nxT (x)(1)holds for all x ∈ R.Obviously, a (1, 0)-Jordan centralizer is a left Jordan centralizer. Sim-ilarly, a (0, 1)-Jordan centralizer is a right Jordan centralizer. In the casewhen m = n = 1 we have the relation2T (x2) = T (x)x+ xT (x), x ∈ R.Vukman [6] has proved that every additive mapping T : R→ R, where R isa 2-torsion free semiprime ring, satisfying the relation above, is a two-sidedcentralizer.Motivated by these results, we introduce the following definition.Definition 2. Let m ≥ 0, n ≥ 0 with m + n 6= 0 be some fixed integersand R an arbitrary ring. An additive mapping T : R→ R is called a gener-alized (m,n)-Jordan centralizer if there exists an (m,n)-Jordan centralizerT0 : R→ R such that(2) (m+ n)T (x2) = mT (x)x+ nxT0(x)holds for all x ∈ R.Similar as above, a generalized (1, 0)-Jordan centralizer is a left Jordancentralizer.In [8], Vukman proved that on a prime ring with a nonzero center Z(R)and char(R) 6= 6mn(m + n) every (m,n)-Jordan centralizer is a two-sidedcentralizer. The natural question here is whether an analogue holds true forgeneralized (m,n)-Jordan centralizers. Theorem 1 answers this question inthe affirmative.Theorem 1. Let m ≥ 1, n ≥ 1 be some fixed integers, let R be a primering with char(R) 6= 6mn(m + n)(m + 2n), and let T : R → R be a gener-alized (m,n)-Jordan centralizer. If Z(R) is nonzero, then T is a two-sidedcentralizer.A note on generalized (m,n)-Jordan centralizers 259In the proof of Theorem 1 we will need the next lemma.Lemma 1. Let m ≥ 0, n ≥ 0 with m + n 6= 0 be some fixed integers, letR be a ring, and let T : R → R be a generalized (m,n)-Jordan centralizer.Then(3) 2(m+ n)2T (xyx) = mnT (x)xy +m(2m+ n)T (x)yx−mnT (y)x2+ 2mnxT0(y)x−mnx2T0(y) + n(m+ 2n)xyT0(x) +mnyxT0(x)for all x, y ∈ R.Proof. If we linearize the relation (2), we get(m+ n)T (xy + yx) = mT (x)y +mT (y)x+ nxT0(y) + nyT0(x)(4)for all x, y ∈ R. Similarly, if we linearize the relation (1), we get(m+ n)T0(xy + yx) = mT0(x)y +mT0(y)x+ nxT0(y) + nyT0(x)(5)for all x, y ∈ R.Now, if we put (m+ n)(xy + yx) instead of y in the relation (4), we get(m+n)2T (x2y+yx2+2xyx) = m(m+n)T (x)(xy+yx)+m(m+n)T (xy+yx)x+ n(m+ n)xT0(xy + yx) + n(m+ n)(xy + yx)T0(x)for all x, y ∈ R. Applying the relation (4) and the relation (5) we obtain2(m+ n)2T (xyx) +m(m+ n)T (x2)y +m(m+ n)T (y)x2+ n(m+ n)x2T0(y) + n(m+ n)yT0(x2)= m(m+ n)T (x)(xy + yx)+m(mT (x)y +mT (y)x+ nxT0(y) + nyT0(x))x+ nx(mT0(x)y +mT0(y)x+ nxT0(y) + nyT0(x))+ n(m+ n)(xy + yx)T0(x)for all x, y ∈ R. Using the relations (2) and (1) we get2(m+ n)2T (xyx) +m(mT (x)x+ nxT0(x))y +m(m+ n)T (y)x2+ n(m+ n)x2T0(y) + ny(mT0(x)x+ nxT0(x))= m(m+n)T (x)(xy+yx)+m(mT (x)y+mT (y)x+nxT0(y)+nyT0(x))x+nx(mT0(x)y+mT0(y)x+nxT0(y)+nyT0(x))+n(m+n)(xy+yx)T0(x)for all x, y ∈ R. Collecting the terms we arrive at2(m+ n)2T (xyx) = mnT (x)xy +m(2m+ n)T (x)yx−mnT (y)x2+ 2mnxT0(y)x−mnx2T0(y) + n(m+ 2n)xyT0(x) +mnyxT0(x)for all x, y ∈ R. This completes the proof.260 A. FošnerProof of Theorem 1. If we put (m+ n)2x2 for x in (2), we get(m+ n)3T (x4) = m(m+ n)2T (x2)x2 + n(m+ n)2x2T0(x2)= m(m+ n)(mT (x)x+ nxT0(x))x2 + n(m+ n)x2(mT0(x)x+ nxT0(x))= m2(m+ n)T (x)x3 +mn(m+ n)xT0(x)x2 +mn(m+ n)x2T0(x)x+ n2(m+ n)x3T0(x).We have therefore(m+ n)3T (x4) = m2(m+ n)T (x)x3 +mn(m+ n)xT0(x)x2(6)+mn(m+ n)x2T0(x)x+ n2(m+ n)x3T0(x)for every x ∈ R. On the other hand, if we put y = (m+n)x2 in the relation(3), we get2(m+ n)3T (x4) = mn(m+ n)T (x)x3 +m(2m+ n)(m+ n)T (x)x3−mn(m+ n)T (x2)x2 + 2mn(m+ n)xT0(x2)x−mn(m+ n)x2T0(x2)+ n(m+ 2n)(m+ n)x3T0(x) +mn(m+ n)x3T0(x)= 2m(m+ n)2T (x)x3 −mn(mT (x)x+ nxT0(x))x2+ 2mnx(mT0(x)x+ nxT0(x))x−mnx2(mT0(x)x+ nxT0(x))+ 2n(m+ n)2x3T0(x)= (2m(m+ n)2 −m2n)T (x)x3 +mn(2m− n)xT0(x)x2+mn(2n−m)x2T0(x)x+ (2n(m+ n)2 −mn2)x3T0(x).We have therefore(7) 2(m+ n)3T (x4) = (2m(m+ n)2 −m2n)T (x)x3+mn(2m−n)xT0(x)x2+mn(2n−m)x2T0(x)x+(2n(m+n)2−mn2)x3T0(x)for every x ∈ R. By comparing (6) and (7) we getmn(m+ 2n)T (x)x3 − 3mn2xT0(x)x2 − 3m2nx2T0(x)x+mn(2m+ n)x3T0(x) = 0for every x ∈ R. The above equality reduces according to the requirementsof the theorem to(m+ 2n)T (x)x3 − 3nxT0(x)x2 − 3mx2T0(x)x+ (2m+ n)x3T0(x) = 0.Since T0 is commuting on R (see the proof of Theorem 2 in [8]), i.e.,[T0(x), x] = T0(x)x− xT0(x) = 0for all x ∈ R, we have(m+ 2n)T (x)x3 − (m+ 2n)T0(x)x3 = 0.A note on generalized (m,n)-Jordan centralizers 261This yields that(T (x)− T0(x))x3 = 0(8)for all x ∈ R.Let F : R→ R be an additive mapping defined by F (x) = T (x)−T0(x),x ∈ R. We would like to show that F (x) = 0 for all x ∈ R. Namely, ifF (x) = T (x) − T0(x) = 0, then T (x) = T0(x) for all x ∈ R, which yieldsthat T is a two-sided centralizer, since T0 is a two-sided centralizer by [8,Theorem 2].We already know that F (x)x3 = 0 for all x ∈ R. Using full linearizationof this relation one obtains∑pi∈S4F (xpi(1))xpi(2)xpi(3)xpi(4) = 0(9)for all x1, x2, x3, x4 ∈ R. Let c be a nonzero central element. Pick any x ∈ Rand set x1 = x2 = x3 = c and x4 = x in (9). We arrive at(αF (c)x+ βF (x)c)c2 = 0,(10)where α = 18 and β = 6. Since R is prime, it follows that αF (c)x+βF (x)c =0 for all x ∈ R. In particular, αF (c)c + βF (c)c = 0, which yields thatF (c) = 0. Therefore from (10), we get F (x) = 0 for all x ∈ R.The above observations lead to the following conjecture.Conjecture 1. Let m ≥ 1, n ≥ 1 be some fixed integers, let R be asemiprime ring with suitable torsion restrictions, and let T : R → R be ageneralized (m,n)-Jordan centralizer. Then T is a two-sided centralizer.At the end, let us also point out, that we do not know yet whether thisconjecture is true even for (m,n)-Jordan centralizers.References[1] K. I. Beidar, W. S. Martindale III, A. V. Mikhalev, Rings with Generalized Identities,Marcel Dekker, Inc. New York (1996).[2] M. Brešar, Jordan derivations on semiprime rings, Proc. Amer. Math. Soc. 104 (1998),1003–1006.[3] M. Brešar, J. Vukman, Jordan derivations on prime rings, Bull. Austral. Math. Soc.37 (1988), 321–322.[4] J. Cusack, Jordan derivations on semiprime rings, Proc. Amer. Math. Soc. 53 (1975),321–324.[5] I. N. Herstein, Jordan derivations of prime rings, Proc. Amer. Math. Soc. 8 (1957),1104–1110.[6] J. Vukman, An identity related to centralizers in semiprime rings, Comment. Math.Univ. Carolin. 40 (1999), 447–456.262 A. Fošner[7] J. Vukman, On (m,n)-Jordan derivations and commutativity of prime rings, Demon-stratio Math. 41 (2008), 773–778.[8] J. Vukman, On (m,n)-Jordan centralizers in rings and algebras, Glas. Mat. 45(1)(2010), 43–53.[9] B. Zalar, On centralizers of semiprime rings, Comment. Math. Univ. Carolin. 32(1991), 609–614.FACULTY OF MANAGEMENTUNIVERSITY OF PRIMORSKACankarjeva 5SI-6104 KOPER, SLOVENIAE-mail: ajda.fosner@fm-kp.siReceived March 3, 2011.

A note on generalized (m,n)-Jordan centralizers

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