Notes on ( α , β )‐derivations

Source 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 .