Some open problems on locally finite or locally nilpotent derivations and ℰ-derivations

Let [Formula: see text] be a commutative ring and [Formula: see text] an [Formula: see text]-algebra. An [Formula: see text]-[Formula: see text]-derivation of [Formula: see text] is an [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 discuss some open problems on whether or not the image of a locally finite (LF) [Formula: see text]-derivation or [Formula: see text]-[Formula: see text]-derivation of [Formula: see text] is a Mathieu subspace [W. Zhao, Generalizations of the image conjecture and the Mathieu conjecture, J. Pure Appl. Algebra 214 (2010) 1200–1216; Mathieu subspaces of associative algebras, J. Algebra 350(2) (2012) 245–272] of [Formula: see text], and whether or not a locally nilpotent (LN) [Formula: see text]-derivation or [Formula: see text]-[Formula: see text]-derivation of [Formula: see text] maps every ideal of [Formula: see text] to a Mathieu subspace of [Formula: see text]. We propose and discuss two conjectures which state that both questions above have positive answers if the base ring [Formula: see text] is a field of characteristic zero. We give some examples to show the necessity of the conditions of the two conjectures, and discuss some positive cases known in the literature. We also show some cases of the two conjectures. In particular, both the conjectures are proved for LF or LN algebraic derivations and [Formula: see text]-[Formula: see text]-derivations of integral domains of characteristic zero.