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., $\delta =\mathrm{Id}_{\mathcal {A}}-\phi$ for some K-algebra endomorphism $\phi$ of ${\mathcal {A}}$). Motivated by the Idempotent conjecture proposed in Zhao (Commun Contemp Math, https://doi.org/10.1142/S0219199717500560, arXiv:1701.05992), 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 (if and) only if $\delta$ is surjective, which generalizes the same result due to Nouazé and Gabriel (J Algebra 6(1):77–99, 1967) and Wright (Illinois J Math 25(3):423–440, 1981) for locally nilpotent K-derivations of commutative K-algebras to locally finite K-derivations and K-${\mathcal {E}}$-derivations $\delta$ of all K-algebras ${\mathcal {A}}$.