幂交换与中心化映射在严格上三角矩阵环上

Prime Gamma Rings with Centralizing and Commuting Generalized Derivations

Let $M$ be a prime $Γ$-ring satisfying a certain assumption and $D$ a nonzero derivation on $M$. Let $f:M\rightarrow M$ be a generalized derivation such that $f$ is centralizing and commuting on a left ideal $J$ of $M$. Then we prove that $M$ is commutative.

$σ$-Mappings of triangular algebras

Let $A$ be an algebra and $σ$ an automorphism of $A$. A linear map $d$ of $A$ is called a $σ$-derivation of $A$ if $d(xy) = d(x)y + σ(x)d(y)$, for all $x, y \in A$. A linear map $D$ is said to be a generalized $σ$-derivation of $A$ if there exists a $σ$-derivation $d$ of $A$ such that $D(xy) = D(x)y + σ(x)d(y)$, for all $x, y \in A$. An additive map $Θ$ of $A$ is $σ$-centralizing if $Θ(x)x - σ(x)Θ(x) \in Z(A)$, for all $x \in A$. In this paper, precise descriptions of generalized $σ$-derivations and $σ$-centralizing maps of triangular algebras are given. Analogues of the so-called commutative …

Centralizing Traces and Lie Triple Isomorphisms on Triangular Algebras

Let $\mathcal{T}$ be a triangular algebra over a commutative ring $\mathcal{R}$ and $\mathcal{Z(T)}$ be the center of $\mathcal{T}$. Suppose that ${\mathfrak q}\colon \mathcal{T}\times \mathcal{T}\longrightarrow \mathcal{T}$ is an $\mathcal{R}$-bilinear mapping and that ${\mathfrak T}_{\mathfrak q}\colon: \mathcal{T}\longrightarrow \mathcal{T}$ is a trace of $\mathfrak{q}$. We describe the form of ${\mathfrak T}_{\mathfrak q}$ satisfying the condition $[{\mathfrak T}_{\mathfrak q}(T), T]\in \mathcal{Z(T)}$ for all $T\in \mathcal{T}$. The question of when ${\mathfrak T}_{\mathfrak q}$ has the p…

三角环上的交换Jordan导子为零

Commuting Jordan derivations on triangular rings are zero

本文证明三角环上的交换Jordan导子恒为零，并推广到满足特定条件的2-无挠半素环或满足条件(P)的环。

$σ$-Biderivations and $σ$-commuting maps of triangular algebras

Let $\A$ be an algebra and $σ$ an automorphism of $\A$. A linear map $d$ of $\A$ is called a $σ$-derivation of $\A$ if $d(xy) = d(x)y + σ(x)d(y)$, for all $x, y \in \A$. A bilinear map $D: \A \times \A \to \A$ is said to be a $σ$-biderivation of $\A$ if it is a $σ$-derivation in each component. An additive map $Θ$ of $\A$ is $σ$-commuting if it satisfies $Θ(x)x - σ(x)Θ(x) = 0$, for all $x \in \A$. In this paper, we introduce the notions of inner and extremal $σ$-biderivations and of proper $σ$-commuting maps. One of our main results states that, under certain assumptions, every $σ$-biderivatio…

关于三角环的交换图

On Commuting graphs of triangular rings

本文研究了非交换环上$2\times 2$上三角矩阵环的交换图，计算了$R$为有限域时的边数、团数及独立数，并证明了当$R=\mathbb{Z}_{n}$非域时该图连通且直径为3。