ON (α,β)-SKEW-COMMUTING AND (α,β)-SKEW-CENTRALIZING MAPS IN RINGS WITH LEFT IDENTITY

Title & Authors
ON (α,β)-SKEW-COMMUTING AND (α,β)-SKEW-CENTRALIZING MAPS IN RINGS WITH LEFT IDENTITY
JUNG, YONG-SOO; CHANG, ICK-SOON;

Abstract
Let R be a ring with left identity. Let G : $\small{R{\times}R{\to}R}$ be a symmetric biadditive mapping and g the trace of G. Let $\small{{\alpha}\;:\;R{\to}R}$ be an endomorphism and $\small{{\beta}\;:\;R{\to}R}$ an epimorphism. In this paper we show the following: (i) Let R be 2-torsion-free. If g is ($\small{{\alpha},{\beta}}$)-skew-commuting on R, then we have G = 0. (ii) If g is ($\small{{\beta},{\beta}}$)-skew-centralizing on R, then g is ($\small{{\beta},{\beta}}$)-commuting on R. (iii) Let $\small{n{\ge}2}$. Let R be (n+1)!-torsion-free. If g is n-($\small{{\alpha},{\beta}}$)-skew-commuting on R, then we have G = 0. (iv) Let R be 6-torsion-free. If g is 2-($\small{{\alpha},{\beta}}$)-commuting on R, then g is ($\small{{\alpha},{\beta}}$)-commuting on R.
Keywords
rings with left identity;
Language
English
Cited by
1.
On Skew Centralizing Traces of Permuting n-Additive Mappings, Kyungpook mathematical journal, 2015, 55, 1, 1
2.
On n-commuting and n-skew-commuting maps with generalized derivations in prime and semiprime rings, Siberian Mathematical Journal, 2011, 52, 3, 516
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