论文标题
实现订单作为团戒指
Realizing orders as group rings
论文作者
论文摘要
订单是一个交换戒指，作为一个阿贝里安群体是有限生成和自由的。如果没有非零的nilpotent元素，则换向环将降低。在本文中，我们使用了一个新工具，即，每个减少订单都具有通用分级的事实，以回答有关将订单作为组环实现的问题。特别是，在环为减少顺序的情况下，我们解决了组环的同构问题。我们证明，任何非零减少的订单$ r $都可以以独特的``最大值''方式写作，直到同构。更确切地说，存在一个$ a $ a $ a $ a $ a $ a $ g $，既可以独特地确定同构，使$ r \ r \ cong a [g] $作为戒指，并且如果$ b $是ring and $ h $是一个组，而$ r \ r \ r \ r \ r \ r \ cong b [像con $ $ j $ j $ j] $ j \ times h \ cong g $作为组。为给定$ r $计算$ a $ a $ g $可以通过不是完全多项式时间的算法来完成的。我们还对$ a $ a $ a $ r $ $ r $的自动形态组进行了描述。
An order is a commutative ring that as an abelian group is finitely generated and free. A commutative ring is reduced if it has no non-zero nilpotent elements. In this paper we use a new tool, namely, the fact that every reduced order has a universal grading, to answer questions about realizing orders as group rings. In particular, we address the Isomorphism Problem for group rings in the case where the ring is a reduced order. We prove that any non-zero reduced order $R$ can be written as a group ring in a unique ``maximal'' way, up to isomorphism. More precisely, there exist a ring $A$ and a finite abelian group $G$, both uniquely determined up to isomorphism, such that $R\cong A[G]$ as rings, and such that if $B$ is a ring and $H$ is a group, then $R\cong B[H]$ as rings if and only if there is a finite abelian group $J$ such that $B\cong A[J]$ as rings and $J\times H\cong G$ as groups. Computing $A$ and $G$ for given $R$ can be done by means of an algorithm that is not quite polynomial-time. We also give a description of the automorphism group of $R$ in terms of $A$ and $G$.
