On rigid derivations in rings - Artemovych, O.D.; Lukashenko, M.P. (Carpathian Mathematical Publications, 2014)
We prove that in a ring R with an identity there exists an element aR and a nonzero derivation dDer R such that ad(a) eq 0. A ring R is said to be a d-rigid ring for some derivation d Der R if d(a)=0 or ad(a) eq 0 for all a R. We study rings with rigid derivations and establish that a commutative Artinian ring R either has a non-rigid derivation or R=R_1… R_n is a ring direct sum of rings R_1,… ,R_n 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 Jacobson radical J(R), for any derivation dDer R such that d(J(R))=0, it follows that d=0_R if and only if the quotient ring R/J(R) is differentially trivial field.