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Idempotents in Intersection of the Kernel and the Image of Locally Finite Derivations and $\mathcal E$-derivations

Let $K$ be a field of characteristic zero, $\mathcal A$ a $K$-algebra and $δ$ a $K$-derivation of $\mathcal A$ or $K$-$\mathcal E$-derivation of $\mathcal A$ (i.e., $δ=\operatorname{Id}_A-ϕ$ for some $K$-algebra endomorphism $ϕ$ of $\mathcal A$). Motivated by the Idempotent conjecture proposed in [Z4], we first show that for every idempotent $e$ lying in both the kernel ${\mathcal A}^δ$ and the image $\operatorname{Im}δ\!:=δ({\mathcal A})$ of $δ$, the principal ideal $(e)\subseteq \operatorname{Im} δ$ if $δ$ is a locally finite $K$-derivation or a locally nilpotent $K$-$\mathcal E$-derivation of $\mathcal A$; and $e{\mathcal A}, {\mathcal A}e \subseteq \operatorname{Im} δ$ if $δ$ 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 $1_{\mathcal A} \in \operatorname{Im} δ$, (if and) only if $δ$ 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 $δ$ of all $K$-algebras $\mathcal A$.