The LNED and LFED Conjectures for Algebraic Algebras

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 $\mathcal A$arefinitedimensionalover$K$\mathrm{.}A$K$-$\mathcal E$-derivationof$\mathcal A$isa$K$-linearmapoftheform$\operatorname{I}-\phi$forsome$K$-algebra endomorphism $\varphi$ of $\mathcal{A}$, where $\mathrm{I}$ 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 $\varphi$ of $\mathcal{A}$, the images of $D$ and $\mathrm{I}-\varphi$ 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 $\delta =\mathrm{I}-\varphi$ with $\varphi$ 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.