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., $\delta ={\mathrm{Id}}_A-\varphi$ for some $K$-algebra endomorphism $\varphi$ 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 $\mathrm{Im}\delta :=\delta (\mathcal{A})$ of $\delta$, the principal ideal $(e)\subseteq \mathrm{Im}\delta$ if $\delta$ is a locally finite $K$-derivation or a locally nilpotent $K$-$\mathcal{E}$-derivation of $\mathcal{A}$; and $e\mathcal{A},\mathcal{A}e\subseteq \mathrm{Im}\delta$ 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 $1_{\mathcal{A}}\in \mathrm{Im}\delta$, (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 $\mathcal{A}$.

16 pages