On rigid derivations in rings

Abstract

We prove that in a ring $R$ with an identity there exists an element $a\in R$ and a nonzero derivation $d\in Der R$ such that $ad(a)\neq 0$. A ring $R$ is said to be a $d$-rigid ring for some derivation $d \in Der R$ if  $d(a)=0$ or $ad(a)\neq 0$ for all $a \in R$. We study rings with rigid derivations and establish that a commutative Artinian ring $R$ either has a non-rigid derivation or $R=R_1\oplus \cdots \oplus R_n$ is a ring direct sum of rings $R_1,\ldots ,R_n$ 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 $d\in Der R$ 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.