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Let $R$ be a commutative ring and $\mathcal A$ an $R$-algebra. An$R$-$\mathcal E$-derivation of $\mathcal A$ is an $R$-linear map of the form$\operatorname{I}-\phi$ for some $R$-algebra endomorphism $\phi$ of $\mathcalA$, where $\operatorname{I}$ denotes the identity map of $\mathcal A$. In thispaper we discuss some open problems on whether or not the image of a locallyfinite $R$-derivation or $R$-$\mathcal E$-derivation of $\mathcal A$ is aMathieu subspace [Z2, Z3] of $\mathcal A$, and whether or not a locallynilpotent $R$-derivation or $R$-$\mathcal E$-derivation of $\mathcal A$ mapsevery ideal of $\mathcal A$ to a Mathieu subspace of $\mathcal A$. We proposeand discuss two conjectures which state that both questions above have positiveanswers if the base ring $R$ is a field of characteristic zero. We give someexamples to show the necessity of the conditions of the two conjectures, anddiscuss some positive cases known in the literature. We also show some cases ofthe two conjectures. In particular, both the conjectures are proved for locallyfinite or locally nilpotent algebraic derivations and $\mathcal E$-derivationsof integral domains of characteristic zero.

Some Open Problems on Locally Finite or Locally Nilpotent Derivations and ${\mathcal E}$-Derivations