Geodesic
Jordan *-Derivations of Finite-DimensionalSemiprime Algebras
Canadian mathematical bulletin, Tome 57 (2014) no. 1, pp. 51-60

Voir la notice de l'article provenant de la source Cambridge University Press

In this paper, we characterize Jordan $*$ -derivations of a 2-torsion free, finite-dimensional semiprime algebra $R$ with involution $*$ . To be precise, we prove the following. Let $\delta :\,R\,\to \,R$ be a Jordan $*$ -derivation. Then there exists a $*$ -algebra decomposition $R\,=\,U\,\oplus \,V$ such that both $U$ and $V$ are invariant under $\delta $ . Moreover, $*$ is the identity map of $U$ and $\delta {{|}_{U}}$ is a derivation, and the Jordan $*$ -derivation $\delta {{|}_{V}}$ is inner. We also prove the following. Let $R$ be a noncommutative, centrally closed prime algebra with involution $*$ , char $R\,\ne \,2$ , and let $\delta $ be a nonzero Jordan $*$ -derivation of $R$ . If $\delta $ is an elementary operator of $R$ , then ${{\dim}_{C}}\,R\,<\,\infty $ and $\delta $ is inner.
DOI : 10.4153/CMB-2012-024-2
Mots-clés : 16W10, 16N60, 16W25, semiprime algebra, involution, (inner) Jordan *-derivation, elementary operator 
Fošner, Ajda; Lee, Tsiu-Kwen. Jordan *-Derivations of Finite-DimensionalSemiprime Algebras. Canadian mathematical bulletin, Tome 57 (2014) no. 1, pp. 51-60. doi: 10.4153/CMB-2012-024-2
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[1] [1] Bergen, J., Derivations in prime rings. Canad. Math. Bull. 26 (1983), no. 3, 267–270. Google Scholar | DOI

[2] [2] Brešar, M. and Vukman, J., On some additive mappings in rings with involution. Aequationes Math. 38 (1989), no. 2–3, 178–185. Google Scholar | DOI

[3] [3] Brešar, M. and Zalar, B., On the structure of Jordan *-derivations. Colloq. Math. 63 (1992), no. 2, 163–171. Google Scholar

[4] [4] Chuang, C.-L., Fošner, A., and Lee, T.-K., Jordan *-derivations of locally matrix rings. Algebr. Represent. Theory, published online December 15, 2011, Google Scholar | DOI

[5] [5] Herstein, I. N., Topics in ring theory. The University of Chicago Press, Chicago, Ill.-London, 1969. Google Scholar

[6] [6] Jacobson, N., PI-algebras. An introduction. Lecture Notes in Mathematics, 441, Springer-Verlag, Berlin-New York, 1975. Google Scholar

[7] [7] Jacobson, N. and Rickart, C. E., Jordan homomorphisms of rings. Trans. Amer. Math. Soc. 69 (1950), 479–502. Google Scholar

[8] [8] Lee, T.-K., Finiteness properties of differential polynomials. Linear Algebra Appl. 430 (2009), no. 8–9, 2030–2041. Google Scholar | DOI

[9] [9] Martindale, W. S. III, Prime rings satisfying a generalized polynomial identity. J. Algebra 12 (1969), 576–584. Google Scholar | DOI

[10] [10] Šemrl, P., Jordan*-derivations on standard operator algebras. Proc. Amer. Math. Soc. 120 (1994), no. 2, 515–518. Google Scholar

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