A ring has unbounded generating number (UGN) if, for every positive integer , there is no -module epimorphism . For a ring graded by a group such that the base ring has UGN, we identify several sets of conditions under which must also have UGN. The most important of these are: (1) is amenable, and there is a positive integer such that, for every , as -modules for some ; (2) is supramenable, and there is a positive integer such that, for every , as -modules for some . 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 , the smallest positive integer such that there is an -module epimorphism is called the generating number of , denoted . If has UGN, then we define . We describe several classes of examples of a ring graded by an amenable group such that .

中文翻译：

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生成按服从组和超级组分级的环数

戒指具有无界生成数( UGN ) 如果对于每个正整数，没有- 模块外同态。对于戒指按小组评分使得基环有UGN，我们确定了几组条件，在这些条件下还必须有 UGN。其中最重要的是：（1）是可以接受的，并且有一个正整数这样，对于每个,作为-一些模块; (2)是不可超越的，并且有一个正整数这样，对于每个,作为-一些模块。一对条件 (1) 导致了群顺应性属性的三种不同的环理论特征。我们还考虑没有 UGN 的戒指；对于这样的一枚戒指, 最小的正整数使得有一个- 模块外同态称为生成数，表示为。如果有UGN，那么我们定义。我们描述几类环的例子由一个顺从的小组评分这样。