Lie rings with derivation

In this thesis, some properties from works which have been done up till now aboutstructure of the Lie ring of derivations of an associative ring R, denoted Der (R),are given.This thesis consists of five chapters.In the first chapter, subject of the thesis is introduced and a short summary ofworks which are related this subject is shortly given. In the second chapter, somebasic definitions and properties are mentioned to make easy understanding of thethesis.In the third chapter, the first section of Herstein's book which is publishedin 1969 is taken essentially to understand the structure of the Lie ring and workswhich are related with this subject and some of works about Lie and Jordanstructures in simple rings are gathered.In the fourth chapter, the properties of Lie ring Der (R) of derivations of Rare investigated for the cases where R is commutative and non-commutative seperately, the theorems which are expressed the Lie ring Der (R) under whichconditions is prime are given. Unless R is commutative, the primeness of Der (R)is investigated by approach is via the study of the structure of I (R), the Lie ringof inner derivations of the prime ring R. In the commutative case, the approachis via the Lie structure of the Lie ring R-delta of all derivations of the form r-delta wherer is in R and delta is a given derivation of R. The properties of R-delta are worked indetail and it's seen that R-delta and I (R) have similar properties. Furthermore, thehypothesis of primeness on R which is a Noetherian ring of characteristic not 2with identity is weakened to delta-primeness and it is proved that R-deltais a prime Lie ring.In the fifth chapter, a theorem is presented in the light of the theorems given inthe previous chapter by taking R only as a commutative ring. In this theorem,the primeness of the Lie ring R-delta is proved without assuming any furthercondition except from the delta-primeness of R which is a 2-torsion free ring withidentity.