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Lie maps on alternative rings

  • Bruno Leonardo Macedo FerreiraEmail author
  • Henrique GuzzoJr.
Article

Abstract

Let \({\mathfrak {R}}\, \) be an alternative ring containing a nontrivial idempotent and \({\mathfrak {R}}\, '\) be another alternative ring. Suppose that a bijective mapping \(\varphi : {\mathfrak {R}}\, \rightarrow {\mathfrak {R}}\, '\) is a Lie multiplicative mapping and \({\mathfrak {D}}\, \) is a Lie triple derivable multiplicative mapping from \({\mathfrak {R}}\, \) into \({\mathfrak {R}}\, \). Under a mild condition on \({\mathfrak {R}}\, \), we prove that \(\varphi \) and \({\mathfrak {D}}\, \) are almost additive, that is, \(\varphi (a + b) - \varphi (a) - \varphi (b) \in \mathcal {Z}({\mathfrak {R}}\, ')\) and \({\mathfrak {D}}\, (a+b) - {\mathfrak {D}}\, (a) - {\mathfrak {D}}\, (b) \in \mathcal {Z}({\mathfrak {R}}\, )\) for all \(a,b \in {\mathfrak {R}}\, \), where \(\mathcal {Z}({\mathfrak {R}}\, ')\) is the commutative centre of \({\mathfrak {R}}\, '\) and \(\mathcal {Z}({\mathfrak {R}}\, )\) is the commutative centre of \({\mathfrak {R}}\, \). As applications, we show that every Lie multiplicative bijective mapping and Lie triple derivable multiplicative mapping on prime alternative rings are almost additive.

Keywords

Additivity Lie multiplicative maps Lie triple derivable maps Prime alternative rings 

Mathematics Subject Classification

17A36 17D05 

Notes

References

  1. 1.
    Changjing, L., Quanyuan, C.: Additivity of Lie multiplicative mappings on rings[J]. Adv. Math. (China) 44(7), 15038 (2015)Google Scholar
  2. 2.
    Changjing, L., Xiaochun, F., Fangyan, L., Ting, W.: Lie triple derivable mappings on rings. Commun. Algebra 42, 2510–2527 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ferreira, J.C.M., Ferreira, B.L.M.: Additivity of \(n\)-multiplicative maps on alternative rings. Commun. Algebra 44, 1557–1568 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ferreira, R.N., Ferreira, B.L.M.: Jordan triple derivation on alternative rings. Proyecciones J. Math. 37, 171–180 (2018)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hentzel, I.R., Kleinfeld, E., Smith, H.F.: Alternative rings with idempotent. J. Algebra 64, 325–335 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Martindale III, W.S.: When are multiplicative mappings additive? Proc. Am. Math. Soc. 21, 695–698 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Slater, M.: Prime alternative rings, I. J. Algebra 15, 229–243 (1970)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Unione Matematica Italiana 2019

Authors and Affiliations

  • Bruno Leonardo Macedo Ferreira
    • 1
    Email author
  • Henrique GuzzoJr.
    • 2
  1. 1.Federal Technological University of ParanáGuarapuavaBrazil
  2. 2.Institute of Mathematics and StatisticsUniversity of São PauloSão PauloBrazil

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