Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-23T02:38:51.232Z Has data issue: false hasContentIssue false
  • English
  • Français

A NOTE ON $(m,n)$-JORDAN DERIVATIONS OF RINGS AND BANACH ALGEBRAS

Part of: Topological (rings and) algebras with an involution Radicals and radical properties of rings

Published online by Cambridge University Press:  28 October 2015

IRENA KOSI-ULBL  and
JOSO VUKMAN
IRENA KOSI-ULBL*
Affiliation:
Faculty of Mechanical Engineering, University of Maribor, Smetanova 17, 2000 Maribor, Slovenia email [email protected]
JOSO VUKMAN
Affiliation:
Department of Mathematics and Computer Science, Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, 2000 Maribor, Slovenia email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we prove the following result: let $m,n\geq 1$ be distinct integers, let $R$ be an $mn(m+n)|m-n|$-torsion free semiprime ring and let $D:R\rightarrow R$ be an $(m,n)$-Jordan derivation, that is an additive mapping satisfying the relation $(m+n)D(x^{2})=2mD(x)x+2nxD(x)$ for $x\in R$. Then $D$ is a derivation which maps $R$ into its centre.

Keywords

prime ringsemiprime ringBanach algebraderivationJordan derivationleft derivationleft Jordan derivation(m, n)-Jordan derivation

MSC classification

Primary: 16N60: Prime and semiprime rings
Secondary: 46K15: Hilbert algebras
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 93 , Issue 2 , April 2016 , pp. 231 - 237
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

Beidar, K. I., Brešar, M., Chebotar, M. A. and Martindale III, W. S., ‘On Herstein’s Lie map conjectures II’, J. Algebra 238 (2001), 239–264.CrossRefGoogle Scholar
Brešar, M., ‘Jordan derivations on semiprime rings’, Proc. Amer. Math. Soc. 104 (1988), 1003–1006.CrossRefGoogle Scholar
Brešar, M., ‘On a generalization of the notion of centralizing mapping’, Proc. Amer. Math. Soc. 114 (1992), 641–649.Google Scholar
Brešar, M., ‘Functional identities: a survey’, Contemp. Math. 259 (2000), 93–109.Google Scholar
Brešar, M., Chebotar, M. A. and Martindale III, W. S., Functional Identities (Birkhauser, Basel, 2007).Google Scholar
Brešar, M. and Vukman, J., ‘Jordan derivations on prime rings’, Bull. Aust. Math. Soc. 37 (1988), 321–322.CrossRefGoogle Scholar
Brešar, M. and Vukman, J., ‘On left derivations and related mappings’, Proc. Amer. Math. Soc. 110 (1990), 7–16.CrossRefGoogle Scholar
Brešar, M. and Vukman, J., ‘Jordan (𝜃, 𝜑)-derivations’, Glas. Mat. 26 (1991), 83–88.Google Scholar
Cusack, J., ‘Jordan derivations on rings’, Proc. Amer. Math. Soc. 53 (1975), 321–324.Google Scholar
Deng, Q., ‘On Jordan left derivations’, Math. J. Okayama Univ. 34 (1992), 145–147.Google Scholar
Fošner, M., Širovnik, N. and Vukman, J., ‘A result related to Herstein theorem’, Bull. Malays. Math. Soc., to appear; doi:10.1007/s40840-015-0196-z.Google Scholar
Fošner, M., Ur Rehmann, N. and Vukman, J., ‘An Engel condition with an additive mapping in semiprime rings’, Proc. Indian Acad. Sci. Math. Sci. 124 (2014), 497–500.Google Scholar
Fošner, M. and Vukman, J., ‘On some functional equations arising from (m, n)-Jordan derivations and commutativity of prime rings’, Rocky Mountain J. Math. 42 (2012), 1153–1168.Google Scholar
Herstein, I. N., ‘Jordan derivations of prime rings’, Proc. Amer. Math. Soc. 8 (1957), 1104–1110.CrossRefGoogle Scholar
Johnson, B. E. and Sinclair, A. M., ‘Continuity of derivations and a problem of Kaplansky’, Amer. J. Math. 90 (1968), 1067–1073.Google Scholar
Posner, E. C., ‘Derivations in prime rings’, Proc. Amer. Math. Soc. 8 (1957), 1093–1100.Google Scholar
Sinclair, A. M., ‘Continuity of derivations on Banach algebras’, Proc. Amer. Math. Soc. 20 (1969), 166–170.Google Scholar
Singer, I. M. and Wermer, J., ‘Derivations on commutative normed algebras’, Math. Ann. 129 (1955), 260–264.Google Scholar
Thomas, M. P., ‘The image of a derivation is contained in the radical’, Ann. of Math. (2) 128 (1988), 435–460.CrossRefGoogle Scholar
Vukman, J., ‘Centralizers on semiprime rings’, Comment. Math. Univ. Carolin. 42 (2001), 237–245.Google Scholar
Vukman, J., ‘On left Jordan derivations of rings and Banach algebras’, Aequationes Math. 75 (2008), 260–266.Google Scholar
Vukman, J., ‘On (m, n)-Jordan derivations and commutativity of prime rings’, Demonstratio Math. 19 (2008), 774–778.Google Scholar
Vukman, J. and Kosi-Ulbl, I., ‘On some equations related to derivations in rings’, Int. J. Math. Sci. (2005), 2703–2710.Google Scholar