ON RIGID DERIVATIONS IN RINGS

We prove that in a ring R with an identity there exists an element a is an element of R and a nonzero derivation d is an element of Der R such that ad(a) not equal 0. A ring R is said to be a d-rigid ring for some derivation d is an element of Der R if d(a) = 0 or ad(a) not equal 0 for all a is an element of 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 circle plus . . . circle plus 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 d is an element of Der 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.