  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 $\mathcal
A$, where $\operatorname{I}$ denotes the identity map of $\mathcal A$. 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.
