Notes on ( α, β ) -derivations

Neşet Aydin

International Journal of Mathematics and Mathematical Sciences, 1997, vol. 20, 1-4

Abstract:

Let R be a prime ring of characteristic not 2 , U a nonzero ideal of R and 0 ≠ d a ( α , β ) -derivation of R where α and β are automorphisms of R . i) [ d ( U ) , a ] = 0 then a ∈ Z ii) For a , b ∈ R , the following conditions are equivalent (I) α ( a ) d ( x ) = d ( x ) β ( b ) , for all x ∈ U (II) Either α ( a ) = β ( b ) ∈ C R ( d ( U ) ) or C R ( a ) = C R ( b ) = R ′ and a [ a , x ] = [ a , x ] b (or a [ b , x ] = [ b , x ] b ) for all x ∈ U . Let R be a 2 -torsion free semiprime ring and U be a nonzero ideal of R iii) Let d be a ( α , β ) -derivation of R and g be a ( γ , δ ) -derivation of R . Suppose that d g is a ( α γ , β δ ) -derivation and g commutes both γ and δ then g ( x ) U α − 1 d ( y ) = 0 , for all x , y ∈ U iv) Let Ann ( U ) = 0 and d be an ( α , β ) -derivation of R and g be a ( λ , δ ) -derivation of R such that g commutes both γ , and δ . If for all x , y ∈ U , β − 1 ( d ( x ) ) U g ( y ) = 0 = g ( x ) U α − 1 ( d ( y ) ) then d g is a ( α γ , β δ ) -derivation on R .

Date: 1997
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DOI: 10.1155/S0161171297001105

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