ON DEPENDENT ELEMENTS AND FREE ACTION OF DERIVATIONS IN SEMIPRIME ?-RINGS Page No: 2043-2047

Nishteman N Suliman and Abdul-Rahman H Majeed

Keywords: Prime ?-ring, semiprime ?-ring, derivation, dependent element free action.

Abstract: In this paper we characterize a dependent element of a derivation on a semiprime ?-ring M. It is shown that the dependent elements of a derivation d of a semiprime ?-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 ?-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 ?- ring M is free action on V. Furthermore, some of results of free action for several mappings in prime and semiprime ?-rings are given. 2000 Mathematics Subject Classification: 16W25, 16W10. 16N16.



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