Lukashenko M. Derivations in rings and semiprimenes

The thesis is devoted to research the derivations of associative rings. Some properties of the rings with all derivations nilpotent of index <2 were studied there. We study derivations that acts as a homomorphism or an antihomomorphism on a differentially semiprime ring (respectively on its nonzero differential right ideal). Let R be a ring with an identity, U a nonzero right d-ideal and d є Der R. There were proved that if R is d-semiprime and d is a homomorphism of R (respectively acts as a homomorphism on U), then d=0. If R is d-prime and d acts as an antihomomorphism on U, then d=0. We introduce the notion of a rigid derivation, investigate its properties and characterize the commutative Artinian ring R with the rigid derivations. In the thesis were proved that in a ring R with an identity there exists an element a є R and a nonzero derivation d such that ad(a) is nonzero. A ring R is said to be a d-rigid ring for some derivation d if d(a)=0 or ad(a) is nonzero for all elements from R. In the work were established that a commutative Artinian ring R either has a non-rigid derivation or is a ring direct sum of rings every of which is a field or a differentially trivial v-ring. The proof of this result is based on the fact that in a local ring R with the nonzero left T-nilpotent Jacobson radical J(R), for any derivation d such that d(J(R))=0, it follows that d=0 if and only if the quotient ring R/J(R) is a differentially trivial field. In the thesis obtained a characterization of commutative rings R with a nonzero derivation d such that d(x)=0 or d(x) is regular for all x є R and a characterization of a structure of the right Goldie rings R which have a nonidentity automorphism v such that x-v(x)=0 or is regular for every x є R. There were also proved that if a right Goldie ring has a such automorphism v, then it is a semiprime ring with the classical right ring of quotients Q which is either a division ring T, or the ring direct sum Q=T+T, or the ring of 2 * 2 matrices over a division ring T. For a 2-torsion free associative rings R there were established the relationships between the properties of R, the associated Jordan ring RJ, the associated left Jordan ring RlJ and the associated Lie ring RL.