Lie maps on alternative rings

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.

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Correspondence to Bruno Leonardo Macedo Ferreira.

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Ferreira, B.L.M., Guzzo, H. Lie maps on alternative rings. Boll Unione Mat Ital 13, 181–192 (2020). https://doi.org/10.1007/s40574-019-00213-9

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Keywords

  • Additivity
  • Lie multiplicative maps
  • Lie triple derivable maps
  • Prime alternative rings

Mathematics Subject Classification

  • 17A36
  • 17D05