Artemovych O.d., Lukashenko M.p. on Rigid Derivations in Rings
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چکیده
We prove that in a ring Rwith an identity there exists an element a ∈ R and a nonzero derivation d ∈ Der R such that ad(a) 6= 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) 6= 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 = R1 ⊕ · · · ⊕ Rn is a ring direct sum of rings R1, . . . , Rn 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 ∈ Der R such that d(J(R)) = 0, it follows that d = 0R if and only if the quotient ring R/J(R) is differentially trivial field.
منابع مشابه
Differentially trivial left Noetherian rings
We characterize left Noetherian rings which have only trivial derivations.
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