论文标题
产生由可融合和可超过群体分级的戒指数量
Generating numbers of rings graded by amenable and supramenable groups
论文作者
论文摘要
一个环$ r $具有{\ it无界的生成号}（ugn），如果对于每个正整数$ n $，则没有$ r $ -module Emimorlism $ r^n \ to r^{n+1} $。对于环$ r = \ bigoplus_ {g \ in g} r_g $由组$ g $分级的r_g $，以使基本环$ r_1 $具有UGN，我们确定了$ r $也必须具有ugn的几套条件。其中最重要的是：（1）$ g $是可正常的，并且有一个正整数$ r $，这样，对于g $中的每一个$ g \，$ r_g \ cong（r_1）^i $ as as $ r_1 $ -mmodules，用于某些$ i = 1，\ dots，r $; （2）$ g $是可超过的，并且有一个正整数$ r $，这样，对于g $中的每$ g \，$ r_g \ cong（r_1）^i $ as $ r_1 $ - modules for Some $ i = 0，\ dots，r $。两对条件（1）导致三种不同的环理论特征对组的合理性。 我们还认为没有UGN的戒指；对于这样的环$ r $，是最小的正整数$ n $，因此有一个$ r $ - 模块的表达$ r^n \ to r^{n+1} $称为$ r $的{\ it生成号}，表示为$ {\ rm gn}（r）$。如果$ r $具有ugn，则我们定义$ {\ rm gn}（r）：= \ aleph_0 $。我们描述了一个由不友善组$ g $分级的环$ r $的几类示例，以便$ {\ rm gn}（r）\ neq {\ rm gn}（r_1）$。
A ring $R$ has {\it unbounded generating number} (UGN) if, for every positive integer $n$, there is no $R$-module epimorphism $R^n\to R^{n+1}$. For a ring $R=\bigoplus_{g\in G} R_g$ graded by a group $G$ such that the base ring $R_1$ has UGN, we identify several sets of conditions under which $R$ must also have UGN. The most important of these are: (1) $G$ is amenable, and there is a positive integer $r$ such that, for every $g\in G$, $R_g\cong (R_1)^i$ as $R_1$-modules for some $i=1,\dots,r$; (2) $G$ is supramenable, and there is a positive integer $r$ such that, for every $g\in G$, $R_g\cong (R_1)^i$ as $R_1$-modules for some $i=0,\dots,r$. The pair of conditions (1) leads to three different ring-theoretic characterizations of the property of amenability for groups. We also consider rings that do not have UGN; for such a ring $R$, the smallest positive integer $n$ such that there is an $R$-module epimorphism $R^n\to R^{n+1}$ is called the {\it generating number} of $R$, denoted ${\rm gn}(R)$. If $R$ has UGN, then we define ${\rm gn}(R):=\aleph_0$. We describe several classes of examples of a ring $R$ graded by an amenable group $G$ such that ${\rm gn}(R)\neq {\rm gn}(R_1)$.
