ON DEPENDENT ELEMENTS AND FREE ACTION OF DERIVATIONS IN SEMIPRIME ?-RINGS

摘要

In this paper we characterize a dependent element of a derivation on a semiprime Γ-ring M. It is shown that the dependent elements of a derivation d of a semiprime Γ-rings M are central, and it is proved that if a is a dependent element of a derivation d of M, then there exist ideals U and V of the semiprime Γ-ring M such that (i)U . V is an essential ideal of M. (ii) d =0 on U, and d(V) .V and (iii) the derivation d of the semiprime Γ- ring M is free action on V. Furthermore, some of results of free action for several mappings in prime and semiprime Γ-rings are given.