Related papers: The LNED and LFED Conjectures for Algebraic Algebr…

Let $K$ be a field of characteristic zero and $x$ a free variable. A $K$-$\mathcal E$-derivation of $K[x]$ is a $K$-linear map of the form $\operatorname{I}-\phi$ for some $K$-algebra endomorphism $\phi$ of $K[x]$, where $\operatorname{I}$…

Let $R$ be an integral domain of characteristic zero, $x=(x_1, x_2, ..., x_n)$ $n$ commutative free variables, and ${\mathcal A}_n:=R[x^{-1}, x]$, i.e., the Laurent polynomial algebra in $x$ over $R$. In this paper we first classify all…

Let $K$ be an algebraically closed field of characteristic zero, $\delta$ a nonzero $\mathcal{E}$-derivation of $K[x]$. We first prove that $\operatorname{Im}\delta$ is a Mathieu-Zhao space of $K[x]$ in some cases. Then we prove that LFED…

Let $K$ be an algebraically closed field of characteristic zero and $A$ an integral $K$-domain. The Lie algebra $Der_{K}(A)$ of all $K$-derivations of $A$ contains the set $LND(A)$ of all locally nilpotent derivations. The structure of…

This paper presents an $\mathcal{E}$-derivation analogue of a result on derivations due to van den Essen, Wright and Zhao. We prove that the image of a locally finite $K$-$\mathcal{E}$-derivation of polynomial algebras in two variables over…

Let $K$ be a field of characteristic zero, $\mathcal A$ a $K$-algebra and $\delta$ a $K$-derivation of $\mathcal A$ or $K$-$\mathcal E$-derivation of $\mathcal A$ (i.e., $\delta=\operatorname{Id}_A-\phi$ for some $K$-algebra endomorphism…

Some cases of the LFED Conjecture, proposed by the second author [Z3], for certain integral domains are proved. In particular, the LFED Conjecture is completely established for the field of fractions $k(x)$ of the polynomial algebra $k[x]$,…

Let $\mathbb K$ be a field of characteristic zero and $A$ an integral domain over $\mathbb K.$ The Lie algebra $\Der_{\mathbb K} A$ of all $\mathbb K$-derivations of $A$ carries very important information about the algebra $A.$ This Lie…

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…

We prove that every local derivation on a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic zero is a derivation. We also give examples of finite-dimensional nilpotent Lie algebras $\mathcal{L}$…

We describe derivations and automorphisms of infinite tensor products of matrix algebras. Using this description we show that for a countable--dimensional locally matrix algebra $A$ over a field $\mathbb{F}$ the dimension of the Lie algebra…

We study in this paper Hida's p-adic Hecke algebra for GL_n over a CM field F. Hida has made a conjecture about the dimension of these Hecke algebras, which he calls the non-abelian Leopoldt conjecture, and shown that his conjecture in the…

This paper considers K\"{o}the's question of whether every associative locally finite-dimensional (abbr., LFD) central division algebra $R$ over a field $K$ is a normally locally finite (abbr., NLF) algebra over $K$, that is, whether every…

Let K be a field of characteristic zero. We prove that images of a linear K-derivation and a linear K-E-derivation of the ring K[x 1 ,x 2 ,x 3 ] of polynomial in three variables over K are Mathieu-Zhao subspaces, which affirms the LFED…

In this paper I consider locally finite Lie algebras of characteristic zero satisfying the condition that for every finite number of elements $x_{1}, x_{2},..., x_{k}$ of such an algebra $L$ there is finite-dimensional subalgebra $A$ which…

We prove that a linear mapping on the algebra \(\mathfrak{sl}_n\) of all trace zero complex matrices is a local automorphism if and only if it is an automorphism or an anti-automorphism. We also show that a linear mapping on a simple…

Let $K _{m}$ be an $m$-local field with an $m$-th residue field $K _{0}$, for some integer $m > 0$, and let $K/K _{m}$ be a field extension of transcendence degree trd$(K/K _{m}) \le 1$. This paper shows that if $K _{0}$ is a field of…

Let $K[X_d]=K[x_1,\ldots,x_d]$ be the polynomial algebra in $d$ variables over a field $K$ of characteristic 0. The classical theorem of Weitzenb\"ock from 1932 states that for linear locally nilpotent derivations $\delta$ (known as…

The paper is devoted to the study of local derivations and automorphisms of nilpotent Lie algebras. Namely, we proved that nilpotent Lie algebras with indices of nilpotency $3$ and $4$ admit local derivation (local automorphisms) which is…

We study algebra endomorphisms and derivations of some localized down-up algebras $\A$. First, we determine all the algebra endomorphisms of $\A$ under some conditions on $r$ and $s$. We show that each algebra endomorphism of $\A$ is an…