Images of ideals under derivations and ℰ-derivations of univariate polynomial algebras over a field of characteristic zero

Abstract

Let [Formula: see text] be a field of characteristic zero and [Formula: see text] a free variable. A [Formula: see text]-[Formula: see text]-derivation of [Formula: see text] is a [Formula: see text]-linear map of the form [Formula: see text] for some [Formula: see text]-algebra endomorphism [Formula: see text] of [Formula: see text], where [Formula: see text] denotes the identity map of [Formula: see text]. In this paper, we study the image of an ideal of [Formula: see text] under some [Formula: see text]-derivations and [Formula: see text]-[Formula: see text]-derivations of [Formula: see text]. We show that the LFED conjecture proposed in [W. Zhao, Some open problems on locally finite or locally nilpotent derivations and [Formula: see text]-derivations, Commun. Contem. Math. 20(4) (2018) 1750056] holds for all [Formula: see text]-[Formula: see text]-derivations and all locally finite [Formula: see text]-derivations of [Formula: see text]. We also show that the LNED conjecture proposed in [W. Zhao, Some open problems on locally finite or locally nilpotent derivations and [Formula: see text]-derivations, Commun. Contem. Math. 20(4) (2018) 1750056] holds for all locally nilpotent [Formula: see text]-derivations of [Formula: see text], and also for all locally nilpotent [Formula: see text]-[Formula: see text]-derivations of [Formula: see text] and the ideals [Formula: see text] such that either [Formula: see text], or [Formula: see text], or [Formula: see text] has at least one repeated root in the algebraic closure of [Formula: see text]. As a bi-product, the homogeneous Mathieu subspaces (Mathieu–Zhao spaces) of the univariate polynomial algebra over an arbitrary field have also been classified.