有限维冯·诺伊曼代数情形下Artin-Wedderburn定理的自包含证明

Wedderburn-Artin 定理

The Wedderburn-Artin Theorem

本文针对 Wedderburn-Artin 定理，提出一种简短、自足且不依赖模论的证明方法。

Virtually Semisimple Modules and a Generalization of the Wedderburn-Artin Theorem

By any measure, semisimple modules form one of the most important classes of modules and play a distinguished role in the module theory and its applications. One of the most fundamental results in this area is the Wedderburn-Artin theorem. In this paper, we establish natural generalizations of semisimple modules and give a generalization of the Wedderburn-Artin theorem. We study modules in which every submodule is isomorphic to a direct summand and name them {\it virtually semisimple modules}. A module $_RM$ is called {\it completely virtually semisimple} if each submodules of $M$ is a virtual…

Two Generalizations of the Wedderburn-Artin Theorem with Applications

We say that an $R$-module $M$ is {\it virtually simple} if $M\neq (0)$ and $N\cong M$ for every non-zero submodule $N$ of $M$, and {\it virtually semisimple} if each submodule of $M$ is isomorphic to a direct summand of $M$. We carry out a study of virtually semisimple modules and modules which are direct sums of virtually simple modules. Our theory provides two natural generalizations of the Wedderburn-Artin Theorem and an analogous to the classical Krull-Schmidt Theorem. Some applications of these theorems are indicated. For instance, it is shown that the following statements are equivalent …

关于某些Iwasawa代数的总商环的Wedderburn分解 II

On the Wedderburn decomposition of the total ring of quotients of certain Iwasawa algebras II

本文研究了某些Iwasawa代数的总商环的Wedderburn分解，通过群环的分解来全面确定其结构。

Theorems of Burnside and Wedderburn revisited

We approach celebrated theorems of Burnside and Wedderburn via simultaneous triangularization. First, for a general field $F$, we prove that $M_n(F)$ is the only irrreducible subalgebra of triangularizable matrices in $M_n(F)$ provided such a subalgebra exists. This provides a slight generalization of a well-known theorem of Burnside. Next, for a given $n > 1$, we characterize all fields $F$ such that Burnside's Theorem holds in $M_n(F)$, i.e., $M_n(F)$ is the only irreducible subalgebra of itself. In fact, for a subfield $F$ of the center of a division ring $D$, our simple proof of the aforem…

Representation of artinian partially ordered sets over semiartinian Von Neumann regular algebras

If $R$ is a semiartinian Von Neumann regular ring, then the set $\Prim_{R}$ of primitive ideals of $R$, ordered by inclusion, is an artinian poset in which all maximal chains have a greatest element. Moreover, if $\Prim_{R}$ has no infinite antichains, then the lattice $\BL_{2}(R)$ of all ideals of $R$ is anti-isomorphic to the lattice of all upper subsets of $\Prim_{R}$. Since the assignment $U\mapsto r_R(U)$ defines a bijection from any set $\Simp_R$ of representatives of simple right $R$-modules to $\Prim_{R}$, a natural partial order is induced in $\Simp_R$, under which the maximal element…