论文标题
在组环中使用某些应用的限制组限制组
Restricted-finite groups with some applications in group rings
论文作者
论文摘要
我们对$ g $组进行了一项研究，其中任何无限亚组的索引都是有限的。我们称它们为限制的限制组，并表征有限生成的而不是扭转限制组。我们表明，每个无限限制的限制性阿贝利安组都是同构对$ \ mathbb {z} \ times k $或$ \ mathbb {z} _ {p^\ infty} \ times k $，其中$ k $是有限的组，$ p $是$ p $。我们还证明，当$ g = at $时，$ g = $ a $ a $ t $是$ g $的子群体时，$ g $是无限生成的限制性限制的，这样$ a $ a $ a $ as $ as $ a $ as quasicyclic and $ t $是有限的。作为我们结果的应用，我们表明，如果$ g $不是有限的$ g'$的扭转，并且组环$ rg $的最低条件有限，那么$ r $是半imple戒指，$ g \ cong \ rtimes \ rtimes \ rtimes \ mathbb {z} $，如果$ t $是$ r $ $ r $的订单。在某些条件下，相反的情况也是如此
We carry out a study of groups $G$ in which the index of any infinite subgroup is finite. We call them restricted-finite groups and characterize finitely generated not torsion restricted-finite groups. We show that every infinite restricted-finite abelian group is isomorphic to $\mathbb{Z}\times K$ or $\mathbb{Z}_{p^\infty}\times K$, where $K$ is a finite group and $p$ is a prime number. We also prove that a group $G$ is infinitely generated restricted-finite if and only if $G = AT$, where $A$ and $T$ are subgroups of $G$ such that $A$ is normal quasicyclic and $T$ is finite. As an application of our results, we show that if $G$ is not torsion with finite $G'$ and the group-ring $RG$ has restricted minimum condition then $R$ is a semisimple ring and $G\cong T\rtimes\mathbb{Z} $, where $T$ is finite whose order is unit in $R$. The converse is also true with certain conditions including $G = T\times \mathbb{Z}
