On Automorphisms and Commutativity in Semiprime Rings

Liau, Pao-Kuei ; Liu, Cheng-Kai

Canadian mathematical bulletin, Tome 56 (2013) no. 3, pp. 584-592

Résumé

Let $R$ be a semiprime ring with center $Z\left( R \right)$ . For $x,\,y\,\in \,R$ , we denote by $\left[ x,\,y \right]\,=\,xy\,-\,yx$ the commutator of $x$ and $y$ . If $\sigma $ is a non-identity automorphism of $R$ such that 1 $$\left[ \left[ \cdot \cdot \cdot \,\left[ \left[ \sigma \left( {{x}^{n0}} \right),\,{{x}^{n1}} \right],\,{{x}^{n2}} \right],\cdot \cdot \cdot\right],\,{{x}^{nk}} \right]\,=\,0$$ for all $x\,\in \,R$ , where ${{n}_{0}},\,{{n}_{1}},\,{{n}_{2}},\,...,\,{{n}_{k}}$ are fixed positive integers, then there exists a map $\mu \,:\,R\,\to \,Z\left( R \right)$ such that $\sigma \left( x \right)\,=\,x\,+\,\mu \left( x \right)$ for all $x\,\in \,R$ . In particular, when $R$ is a prime ring, $R$ is commutative.

Détail

DOI : 10.4153/CMB-2011-185-5

Mots-clés : 16N60, 16W20, 16R50, semiprime ring, automorphism, generalized polynomial identity (GPI)

Bibliographie

[1] [1] Albas, E., Argac, N., and De Filippis, V., Generalized derivations with Engel conditions on one-sided ideals. Comm. Algebra 36 (2008), no. 6, 2063–2071. Google Scholar | DOI

[2] [2] Beidar, K. I.,Martindale, W. S. III, and Mikhalev, A. V., Rings with generalized identities. Monographs and Textbooks in Pure and Applied Mathematics, 196, Marcel Dekker, Inc., New York, 1996. Google Scholar

[3] [3] Bell, H. E. and Martindale, W. S. III, Centralizing mappings of semiprime rings. Canad. Math. Bull. 30 (1987), no. 1, 92–101. Google Scholar | DOI

[4] [4] Brešar, M., Centralizing mappings and derivations in prime rings. J. Algebra 156 (1993), no. 2, 385–394. Google Scholar | DOI

[5] [5] Brešar, M., Chebotar, M. A., andW. S.Martindale III, Functional Identities. Frontiers in mathematics. Birkh¨auser Verlag, Basel, 2007. Google Scholar

[6] [6] Chou, M.-C. and Liu, C.-K., An Engel condition with skew derivations. Monatsh. Math. 158 (2009), 259-270. Google Scholar | DOI

[7] [7] Chuang, C.-L., Differential identities with automorphisms and antiautomorphisms. I. J. Algebra 149 (1992), no. 2, 371–404. Google Scholar | DOI

[8] [8] Chuang, C.-L., Differential identities with automorphisms and antiautomorphisms. II. J. Algebra 160 (1993), no. 1, 130–171. Google Scholar | DOI

[9] [9] Chuang, C.-L. and Liu, C.-K., Extended Jacobson density theorem for rings with skew derivations. Comm. Algebra 35 (2007), no. 4, 1391–1413. Google Scholar | DOI

[10] [10] Deng, Q. and Bell, H. E., On derivations and commutativity in semiprime rings. Comm. Algebra 23 (1995), no. 10, 3705–3713. Google Scholar | DOI

[11] [11] Divinsky, N., On commuting automorphisms of rings. Trans. Roy. Soc. Canada. Sect. III. 49 (1955), 19–22. Google Scholar

[12] [12] Erickson, T. S., Martindale, W. S. III, and Osborn, J. M., Prime nonassociative algebras. Pacific J. Math. 60 (1975), 49–63. Google Scholar

[13] [13] De Filippis, V., An Engel condition with generalized derivations on multilinear polynomials. Israel J. Math. 162 (2007), 93–108. Google Scholar | DOI

[14] [14] De Filippis, V., Generalized derivations with Engel condition on multilinear polynomials. Israel J. Math. 171 (2009), 325–348. Google Scholar | DOI

[15] [15] Jacobson, N., Lie algebras. Interscience Tracts in Pure and Applied Mathematics, 10, Interscience Publishers, New York, 1962. Google Scholar

[16] [16] Jacobson, N., Structure of rings. American Mathematical Society Colloquium Publications, 37, American Mathematical Society, Providence, RI, 1964. Google Scholar

[17] [17] Lanski, C., An Engel condition with derivation. Proc. Amer. Math. Soc. 118 (1993), no. 3, 731–734. Google Scholar | DOI

[18] [18] Lanski, C., An Engel condition with derivation for left ideals. Proc. Amer. Math. Soc. 125 (1997), no. 2, 339–345. Google Scholar | DOI

[19] [19] Lee, P.-H. and Wong, T.-L., Derivations cocentralizing Lie ideals. Bull. Inst. Math. Acad. Sinica 23 (1995), no. 1, 1–5. Google Scholar

[20] [20] Lee, P.-H. and YuWang, , Supercentralizing maps in prime superalgebras. Comm. Algebra 37 (2009), 840–854. Google Scholar | DOI

[21] [21] Lee, T.-K., Semiprime rings with hypercentral derivations. Canad. Math. Bull. 38 (1995), no. 4, 445–449. Google Scholar | DOI

[22] [22] Lee, T.-K. and Wong, T.-L., On certain subgroups of prime rings with automorphisms. Comm. Algebra 30 (2002), no. 10, 4997–5009. Google Scholar | DOI

[23] [23] Lee, T.-K. and K.-Liu, S., The Skolem-Noether theorem for semiprime rings satisfying a strict identity. Comm. Algebra 35 (2007), no. 6, 1949–1955. Google Scholar | DOI

[24] [24] Lee, T.-K.and Zhou, Y., An identity with generalized derivations. J. Algebra Appl. 9 (2009), no. 3, 307–317. Google Scholar | DOI

[25] [25] Liu, C.-K., Derivations with Engel and annihilator conditions on multilinear polynomials. Comm. Algebra 33 (2005), no. 3, 719–725. Google Scholar | DOI

[26] [26] Liu, C.-K., Derivations cocentralizing multilinear polynomials on left ideal. Monatsh. Math. 162 (2011), no. 3, 297–311. Google Scholar | DOI

[27] [27] Luh, J., A note on commuting automorphisms of rings. Amer. Math. Monthly 77 (1970), 61–62. Google Scholar | DOI

[28] [28] Martindale, W. S. III, Prime rings satisfying a generalized polynomial identity. J. Algebra 12 (1969), 576–584. Google Scholar | DOI

[29] [29] Mayne, J. H., Centralizing automorphisms of prime rings. Canad. Math. Bull. 19 (1976), no. 1, 113–115. Google Scholar | DOI

[30] [30] Mayne, J. H., Centralizing automorphisms of Lie ideals in prime rings. Canad. Math. Bull. 35 (1992), no. 4, 510–514. Google Scholar | DOI

[31] [31] Vukman, J., Commuting and centralizing mappings in prime rings. Proc. Amer. Math. Soc. 109 (1990), no. 1, 47–52. Google Scholar | DOI

[32] [32] YuWang, , A note on Lie automorphisms, subrings, and Lie ideals of prime rings. Comm. Algebra 33 (2005), no. 11, 4057–4062. Google Scholar | DOI

[33] [33] YuWang, , Power-centralizing automorphisms of Lie ideals in prime rings. Comm. Algebra 34 (2006), no. 2, 609–615. Google Scholar | DOI

[34] [34] YuWang, , Annihilator conditions of derivations on multilinear polynomial. Comm. Algebra 39 (2011), no. 1, 237–246. Google Scholar | DOI

[35] [35] Wong, T.-L., On Lie automorphisms, additive subgroups, and Lie ideals of prime rings. Comm. Algebra 31 (2003), 969–979. Google Scholar | DOI