Aequationes mathematicae

, Volume 85, Issue 3, pp 529–537

# On one sided ideals of a semiprime ring with generalized derivations

Article

## Abstract

Let R be a ring with center Z(R). An additive mapping $${F : R \longrightarrow R}$$ is said to be a generalized derivation on R if there exists a derivation $${d : R \longrightarrow R}$$ such that F(xy) = F(x)y + xd(y), for all $${x, y \in R}$$ (the map d is called the derivation associated with F). Let R be a semiprime ring and U be a nonzero left ideal of R. In the present note we prove that if R admits a generalized derivation F, d is the derivation associated with F such that d(U) ≠ (0) then R contains some nonzero central ideal, if one of the following conditions holds: (1) R is 2-torsion free and $${F(xy) \in Z(R)}$$, for all $${x, y \in U}$$, unless F(U)U = UF(U) = Ud(U) = (0); (2) $${F(xy) \mp yx \in Z(R)}$$, for all $${x,y \in U}$$; (3) $${F(xy) \mp [x,y] \in Z(R)}$$, for all $${x,y \in U}$$; (4) F ≠ 0 and F([x,y]) = 0, for all $${x, y \in U}$$, unless Ud(U) = (0); (5) F ≠ 0 and $${F([x, y]) \in Z(R)}$$, for all $${x, y \in U}$$, unless either d(Z(R))U = (0) or Ud(U) = (0)n.

## Mathematics Subject Classification (2000)

16W25 16W20 16N60

## Keywords

Prime and semiprime ring generalized derivation

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