Some Open Problems on Locally Finite or Locally Nilpotent Derivations and ${\mathcal E}$-Derivations

TL;DR

This paper explores open problems regarding whether locally finite or locally nilpotent derivations and ${ m E}$-derivations on algebras over fields of characteristic zero have images that are Mathieu subspaces, proposing conjectures and providing partial results.

Contribution

The paper introduces two conjectures on the behavior of locally finite and locally nilpotent derivations and ${ m E}$-derivations, and proves these conjectures in specific algebraic cases.

Findings

Conjectures hold for algebraic derivations of integral domains in characteristic zero.

Examples demonstrate the necessity of conditions in the conjectures.

Partial proofs confirm the conjectures for certain derivations and ${ m E}$-derivations.

Abstract

Let $R$ be a commutative ring and $\mathcal A$ an $R$-algebra. An $R$-$\mathcal E$-derivation of $\mathcal A$ is an $R$-linear map of the form $\operatorname{I}-\phi$ for some $R$-algebra endomorphism $\phi$ of $\mathcal A$, where $\operatorname{I}$ denotes the identity map of $\mathcal A$. In this paper we discuss some open problems on whether or not the image of a locally finite $R$-derivation or $R$-$\mathcal E$-derivation of $\mathcal A$ is a Mathieu subspace [Z2, Z3] of $\mathcal A$, and whether or not a locally nilpotent $R$-derivation or $R$-$\mathcal E$-derivation of $\mathcal A$ maps every ideal of $\mathcal A$ to a Mathieu subspace of $\mathcal A$. We propose and discuss two conjectures which state that both questions above have positive answers if the base ring $R$ 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 locally finite or locally nilpotent algebraic derivations and $\mathcal E$-derivations of integral domains of characteristic zero.