Title: Idempotents in Intersection of the Kernel and the Image of Locally Finite Derivations and $\mathcal E$-derivations,
Abstract: Let $K$ be a field of characteristic zero, $\mathcal A$ a $K$-algebra and
$\delta$ a $K$-derivation of $\mathcal A$ or $K$-$\mathcal E$-derivation of
$\mathcal A$ (i.e., $\delta=\operatorname{Id}_A-\phi$ for some $K$-algebra
endomorphism $\phi$ of $\mathcal A$). Motivated by the Idempotent conjecture
proposed in [Z4], we first show that for every idempotent $e$ lying in both the
kernel ${\mathcal A}^\delta$ and the image $\operatorname{Im}\delta \!:=\delta
({\mathcal A})$ of $\delta$, the principal ideal $(e)\subseteq
\operatorname{Im} \delta$ if $\delta$ is a locally finite $K$-derivation or a
locally nilpotent $K$-$\mathcal E$-derivation of $\mathcal A$; and $e{\mathcal
A}, {\mathcal A}e \subseteq \operatorname{Im} \delta$ if $\delta$ is a locally
finite $K$-$\mathcal E$-derivation of $\mathcal A$. Consequently, the
Idempotent conjecture holds for all locally finite $K$-derivations and all
locally nilpotent $K$-$\mathcal E$-derivations of $\mathcal A$. We then show
that $1_{\mathcal A} \in \operatorname{Im} \delta$, (if and) only if $\delta$
is surjective, which generalizes the same result [GN, W] for locally nilpotent
$K$-derivations of commutative $K$-algebras to locally finite $K$-derivations
and $K$-$\mathcal E$-derivations $\delta$ of all $K$-algebras $\mathcal A$.