半artinian Von Neumann正则代数上artinian偏序集的表示
Representation of artinian partially ordered sets over semiartinian Von Neumann regular algebras
摘要：
如果$R$是一个半阿丁冯诺依曼正则环，则$R$的本原理想的集合$\Prim_{R}$，按包含序，是一个阿丁偏序集，其中所有极大链都有一个最大元素。此外，如果$\Prim_{R}$没有无限反链，则$R$的所有理想的格$\BL_{2}(R)$与$\Prim_{R}$的所有上子集的格反同构。由于赋值$U\mapsto r_R(U)$定义了从单右$R$ -模的代表的任何集合$\Simp_R$到$\Prim_{R}$的双射，在$\Simp_R$中诱导了一个自然偏序，在该偏序下，最大元素恰好是那些在各自的自同态除环上是有限维的单右$R$ -模;这些总是$R$ -内射的。给出了一种构造方法，它构造了一个半阿丁单位正则的$D$ -代数$D_I$，它具有以下特征：（a）$\Simp_{D_I}$与$I$序同构;（b）赋值$H\mapsto\Simp_{D_I/H}$实现了从格$\BL_{2}(D_I)$到$\Simp_{D_I}$的所有上子集的格的反同构;（c）$\Simp_{D_I}$的一个非极大元是内射的当且仅当它对应于$I'$的一个元，因此$D_I$是右$V$ -环当且仅当$I' = I$ ;（d）$D_I$是右\emph{and}左$V$ -环当且仅当$I$是反链;（e）如果$I$有有限的对偶Krull长度，则$D_I$是（f）若$I$至多可数且$I' = \vu$，则$D_I$是可数维$D$ -代数.
英文摘要：
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 elements are precisely those simple right $R$-modules which are finite dimensional over the respective endomorphism division rings; these are always $R$-injective. Given any artinian poset $I$ with at least two elements and having a finite cofinal subset, a lower subset $I'\sbs I$ and a field $D$, we present a construction which produces a semiartinian and unit-regular $D$-algebra $D_I$ having the following features: (a) $\Simp_{D_I}$ is order isomorphic to $I$; (b) the assignment $H\mapsto\Simp_{D_I/H}$ realizes an anti-isomorphism from the lattice $\BL_{2}(D_I)$ to the lattice of all upper subsets of $\Simp_{D_I}$; (c) a non-maximal element of $\Simp_{D_I}$ is injective if and only if it corresponds to an element of $I'$, thus $D_I$ is a right $V$-ring if and only if $I' = I$; (d) $D_I$ is a right \emph{and} left $V$-ring if and only if $I$ is an antichain; (e) if $I$ has finite dual Krull length, then $D_I$ is (right and left) hereditary; (f) if $I$ is at most countable and $I' = \vu$, then $D_I$ is a countably dimensional $D$-algebra.
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