# The LNED and LFED Conjectures for Algebraic Algebras

## TL;DR

This paper investigates the LNED and LFED conjectures for algebraic algebras over characteristic zero fields, proving them in various cases including finite dimensional algebras, and explores properties of derivations and automorphisms.

## Contribution

It proves the LNED and LFED conjectures for algebraic algebras, especially finite dimensional ones, and establishes finite extensions of derivations and automorphisms to inner forms.

## Key findings

- Proved LNED conjecture for algebraic algebras.
- Established LFED conjecture for locally finite derivations and certain E-derivations.
- Showed finite extensions of derivations and automorphisms to inner derivations and automorphisms.

## Abstract

Let $K$ be a field of characteristic zero and $\mathcal A$ a $K$-algebra such that all the $K$-subalgebras generated by finitely many elements of $\mathcal A$ are finite dimensional over $K$. A $K$-$\mathcal E$-derivation of $\mathcal A$ is a $K$-linear map of the form $\operatorname{I}-\phi$ for some $K$-algebra endomorphism $\phi$ of $\mathcal A$, where $\operatorname{I}$ denotes the identity map of $\mathcal A$. In this paper we first show that for all locally finite $K$-derivations $D$ and locally finite $K$-algebra automorphisms $\phi$ of $\mathcal A$, the images of $D$ and $\operatorname{I}-\phi$ do not contain any nonzero idempotent of $\mathcal A$. We then use this result to show some cases of the LFED and LNED conjectures proposed in [Z4]. More precisely, We show the LNED conjecture for $\mathcal A$, and the LFED conjecture for all locally finite $K$-derivations of $\mathcal A$ and all locally finite $K$-$\mathcal E$-derivations of the form $\delta=\operatorname{I}-\phi$ with $\phi$ being surjective. In particular, both conjectures are proved for all finite dimensional $K$-algebras. Furthermore, some finite extensions of derivations and automorphism to inner derivations and inner automorphisms, respectively, have also been established. This result is not only crucial in the proofs of the results above, but also interesting on its own right.