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

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 $\mathrm{I}-\varphi$ for some $R$-algebra endomorphism $\varphi$ of $\mathcal A$,where$\operatorname{I}$denotestheidentitymapof$\mathcal A$. In this paper we discuss some open problems on whether or not the image of a locally finite $R$-derivation or $R$-$\mathcal{E}$-derivation of $\mathcal{A}$ is a Mathieu subspace [Z2, Z3] of $\mathcal{A}$, and whether or not a locally nilpotent $R$-derivation or $R$-$\mathcal{E}$-derivation of $\mathcal{A}$ maps every ideal of $\mathcal{A}$ to a Mathieu subspace of $\mathcal{A}$. We propose and discuss two conjectures which state that both questions above have positive answers if the base ring $R$ is a field of characteristic zero. We give some examples to show the necessity of the conditions of the two conjectures, and discuss some positive cases known in the literature. We also show some cases of the two conjectures. In particular, both the conjectures are proved for locally finite or locally nilpotent algebraic derivations and $\mathcal{E}$-derivations of integral domains of characteristic zero.