(ϕ, φ)-derivations on semiprime rings and Banach algebras
Abstract
Let R be a semiprime ring with unity e and φ, ϕ be automorphisms of R. In this paper it is shown that if R satisfies
2D(x
n
) = D(x
n−1
)φ(x) + ϕ(x
n−1
)D(x) + D(x)φ(x
n−1
) + ϕ(x)D(x
n−1
)
for all x ∈ R and some fixed integer n ≥ 2, then D is an (φ, ϕ)-derivation.
Moreover, this result makes it possible to prove that if R admits an additive
mappings D, G : R → R satisfying the relations
2D(x
n
) = D(x
n−1
)φ(x) + ϕ(x
n−1
)G(x) + G(x)φ(x
n−1
) + ϕ(x)G(x
n−1
),
2G(x
n
) = G(x
n−1
)φ(x) + ϕ(x
n−1
)D(x) + D(x)φ(x
n−1
) + ϕ(x)D(x
n−1
),
for all x ∈ R and some fixed integer n ≥ 2, then D and G are (φ, ϕ)-
-derivations under some torsion restriction. Finally, we apply these purely
ring theoretic results to semi-simple Banach algebras.
Domains
Mathematics [math]Origin | Explicit agreement for this submission |
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