Generalized skew derivations on triangular algebras determined by action on zero products

ABSTRACT

For a triangular algebra 𝒜 and an automorphism σ of 𝒜, we describe linear maps F,G:𝒜→𝒜 satisfying F(x)y+σ(x)G(y) = 0 whenever x,y∈𝒜 are such that xy = 0. In particular, when 𝒜 is a zero product determined triangular algebra, maps F and G satisfying the above condition are generalized skew derivations of the form F(x) = F(1)x+D(x) and G(x) = σ(x)G(1)+D(x) for all x∈𝒜, where D:𝒜→𝒜 is a skew derivation. When 𝒜 is not zero product determined, we show that there are also nonstandard solutions for maps F and G.

KEYWORDS:

• Derivation
• generalized derivable mapping at zero point
• generalized skew derivation
• triangular algebra
• zero product determined algebra

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16W25

Acknowledgment

The authors would like to thank the referee for careful reading of the paper.