Let H be a prime ring and U be a nonzero ideal of R. If T is a nontrivial automorphism or derivation of Ft such that uuT — uTu is in the center of R and uT is in U for every u in U, then R is commutative. If R does not have characteristic equal to two, then U need only be a nonzero Jordan ideal

Mayne, Joseph H

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Loyola University ChicagoLoyola eCommonsMathematics and Statistics: Faculty Publicationsand Other Works Faculty Publications10-1982Ideals and Centralizing Mappings in Prime RingsJoseph H. MayneLoyola University Chicago, jmayne@luc.eduThis Article is brought to you for free and open access by the Faculty Publications at Loyola eCommons. It has been accepted for inclusion inMathematics and Statistics: Faculty Publications and Other Works by an authorized administrator of Loyola eCommons. For more information, pleasecontact ecommons@luc.edu.This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 License.© American Mathematical Society, 1982.Recommended CitationMayne, JH. "Ideals and Centralizing Mappings in Prime Rings." Proceedings of the American Mathematical Society 86(2), 1982.PROCEEDINGS of theAMERICAN MATHEMATICAL SOCIETYVolume 86, Number 2, October 1982IDEALS AND CENTRALIZING MAPPINGS IN PRIME RINGSJOSEPH H. MAYNEABSTRACT. Let H be a prime ring and U be a nonzero ideal of R. If T isa nontrivial automorphism or derivation of Ft such that uuT — uTu is in thecenter of R and uT is in U for every u in U, then R is commutative. If R doesnot have characteristic equal to two, then U need only be a nonzero Jordanideal.If R is a ring, a mapping T of R to itself is called centralizing on a subset S ofR if ssT — sTs is in the center of R for every s in S. There has been considerableinterest in centralizing automorphisms and derivations defined on rings. Miers [4]has studied these mappings defined on C '-algebras. In [5] Posner proved thatif a prime ring has a nontrivial centralizing derivation, then the ring must becommutative. The same result was obtained for centralizing automorphisms in[3]. In this paper it is shown that the automorphism or derivation need only becentralizing and invariant on a nonzero ideal in the prime ring in order to ensurethat the ring is commutative. Also, if R is of characteristic not two, then themapping need only be centralizing and invariant on a nonzero Jordan ideal. Forderivations this gives a short proof of a result related to that of Awtar [1, Theorem3]. Awtar proved that if R is a prime ring of characteristic not two with a nontrivialderivation and a nonzero Jordan ideal U such that the derivation is centralizing onU, then U is contained in the center of R.Jeffrey Bergen deserves many thanks for his suggestions concerning the resultsand proofs in this paper.From now on assume that R is a prime ring and let Z be the center of R. Let[x,y] = xy — yx and note the important identity [x,yz] = y[x, z] + [x, y]z. Thefollowing lemmas will be used in the proofs of the main results.LEMMA 1.   If b[a, r]=0 for all r in R, then 6 = 0 or a is in Z.PROOF. Assume that b[a,r] = 0 for all r in R. Replace r by xy to obtainb[a,xy] = bx[a,y] + b[a,x]y = bx[a,y] = 0 for all x and y in R. Since R is prime,b = 0 or [a, y] = 0 for all x in R.LEMMA 2. If D is a derivation ofR such that vP = 0 for all u in a nonzero rightideal U of R, then rD = 0 for all r inR.PROOF. Let u be a nonzero element in U and x be an element in R. Thenux is in U and 0 = (ux)D — uDx + u(xD) = u(xD). Now replace X by sr to obtain0 = u(sr)D — [u(sD)]r + us(rD) = us(rD) for all r and s in R. Since R is prime andu is nonzero, rD = 0 for all r in R.Received by the editors October 21, 1981 and, in revised form, January 12, 1982.1980 Mathematics Subject Classification. Primary 16A70; Secondary 16A12, 16A68, 16A72.Key words and phrases. Ideal, centralizing automorphism, centralizing derivation, prime ring,commutative.© 1982 American Mathematical Society0002-9939/82/0000-0069/S01.T5211License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use212 J. H. MAYNELEMMA 3. IfT is a homomorphism ofR such that uT = u for all u in a nonzeroright ideal U ofR, then rT = r for every r in R.PROOF. Let u be a nonzero element in U and r, s be in R. Since U is aright ideal, us and usr are in U. Then (usr)T = usr = (us)TrT = usrT. Henceus(r — rT) = 0 for all s and r in R. Thus r = rT for all r in R.LEMMA 4. If R contains a nonzero commutative right ideal U, then R must becommutative.PROOF. Let u be in U and assume that u2 is not zero. Such an element existsfor if not, then by a variation of Levitzki's theorem [2, Lemma 1.1], R has a nonzeronilpotent ideal and this is impossible in a prime ring. U is a right ideal and so urand us are in U for every r and s in R. Since U is commutative, u2sr = u(us)r =us(ur) = ur(us) = u(ur)s = u2rs. Hence u2[r, s]=Q for all r and s in R. By Lemma1, every r in R is in Z. Therefore R is commutative.Theorem. Let R be a prime ring and U be a nonzero ideal of R. If R has anontrivial automorphism or derivation T such that uuT — uTu is in the center ofRand uT is in U for every u in U, then R is commutative.PROOF. By Lemma 2 or Lemma 3, T is nontrivial on U. Since U is a nonzeroideal in a prime ring, U is itself a prime ring. U is then commutative by the author'sresult in [3] for automorphisms or by Posner's result [5] for derivations. By Lemma4, R is commutative.COROLLARY. If U is a nonzero Jordan ideal in a prime ring R of characteristicnot two and T is a nontrivial automorphism or derivation ofR which is centralizingand invariant on U, then R is commutative.PROOF. Every nonzero Jordan ideal in a prime ring of characteristic not twocontains a nonzero ideal [2, Theorem 1.1]. Apply the theorem to this ideal.References1. R. Awtar, Lie and Jordan structures in prime rings with derivations, Proc. Amer. Math. Soc. 41(1973), 67-74.2. I. Herstein, Topics in ring theory, Univ. of Chicago Press, Chicago, Illinois, 1969.3. J. Mayne, Centralizing automorphisms of prime rings, Canad. Math. Bull. 19 (1976), 113-115.4. C. R. Miers, Centralizing mappings of operator algebras, J. Algebra 59 (1979), 56-64.5. E. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093-1100.Department  of  Mathematical  Sciences,   Loyola  University,   Chicago,Illinois 60626License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

Ideals and Centralizing Mappings in Prime Rings

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Erratum to \u22Ideals and Centralizing Mappings in Prime Rings\u22

Loyola University Chicago Loyola eCommons Mathematics and Statistics: Faculty Publications and Other Works Faculty Publications 9-1983 Erratum to "Ideals and Centralizing Mappings in Prime Rings" Joseph H. Mayne Loyola University Chicago, jmayne@luc.edu Follow this and additional works at: https://ecommons.luc.edu/math_facpubs  Part of the Mathematics Commons Recommended Citation Mayne, JH. "Erratum to 'Ideals and Centralizing Mappings in Prime Rings.'" Proceedings of the American Mathematical Society 89(1), 1983. This Article is brought to you for free and open access by the Faculty Publications at Loyola eCommons. It has been accepted for inclusion in Mathematics and Statistics: Faculty Publications and Other Works by an authorized administrator of Loyola eCommons. For more information, please contact ecommons@luc.edu. © American Mathematical Society, 1983. proceedings of theamerican mathematical societyVolume 89, Number 1, September 1983ERRATUM TO "IDEALS AND CENTRALIZING MAPPINGSIN PRIME RINGS"JOSEPH H. MAYNEIn the proof of the Theorem we want the automorphism T to induce anautomorphism on the ideal U, but this may not be the case, as pointed out by HisaoTominaga (let R be the polynomial ring over the integers in indeterminates x¡,i' = 0, ± 1, ±2,...,  and let U he the ideal of R generated by xx, x2. If T isdefined by T(x¡) = x,+x, then T(U) C U).To handle this case, letK= ■■■ +T~2(U) + T-\U) + U+ T(U) + T2(U) + ■■■ .A' is a nonzero ideal and T induces an automorphism on K. Now we need to showthat [x, T(x)] is in the center Z of R for every x in K. For x in K,x = Tk^(xk0)) + T^(xH2)) + ---+T^(xkw)where each xk{i) is in U. Letm = absolute value [min{0, k(\), k(2),.. .,k(n)}].Then T"'(x) = y is in U since U is invariant under T. Hence Tm[x, T(x)] —[Tm(x), Tm+\x)\ = [y, T(y)] is in Z because T is centralizing on U. Since T is anautomorphism, [x, T(x)] is in Z for every x in K. Apply the result of Centralizingautomorphisms of prime rings [Canad. Math. Bull. 19 (1976), 113-115] to A".In the Corollary the same problem may occur and can be remedied in theautomorphism case by a similar argument using the fact that the Jordan ideal isinvariant under T. For derivations such that the associative ideal / is not invariant,use K = I + D(I) + D2(I) + •••, which is a nonzero ideal contained in U andinvariant under D.Department of Mathematical Sciences, Loyola University, Chicago, Illinois 60626Joseph H. Mayne, Ideals and centralizing mappings in prime rings, Proc. Amer. Math. Soc. 86 (1982),211-212.Received by the editors April 4, 1983.187License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

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Erratum to  Ideals and Centralizing Mappings in Prime Rings

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Ideals and Centralizing Mappings in Prime Rings

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