阿贝尔群的子群、群环的相关理想及这些理想的子群
Subgroups of an abelian group, related ideals of the group ring, and quotients by those ideals
摘要：
设$RG$是具有单位元的交换环$R$上的阿贝尔群$G$的群环.从$G$的子群到$RG$的非单位理想的注入$Φ$是众所周知的。它由$Φ(N)=I(R,N)RG$定义，其中$I(R,N)$是$RN$的增广理想，并且每个理想$Φ(N)$具有一个性质：$RG/Φ(N)$是同构于$R(G/N)$的$R$ -代数。设$T$是$RG$的非单位理想集.虽然$Φ$的图像是$T$的一个很小的子集，但我们给出了$R$和$G$的条件，使$Φ$的图像在$T$中具有某种分布。在最后一节中，我们给出了$RG$的元素$x$的选择准则，使得对于$G$的子群$N$，$RG/xRG$是同构于$R(G/N)$的$R$ -代数。
英文摘要：
Let $RG$ be the group ring of an abelian group $G$ over a commutative ring $R$ with identity. An injection $Φ$ from the subgroups of $G$ to the non-unit ideals of $RG$ is well-known. It is defined by $Φ(N)=I(R,N)RG$ where $I(R,N)$ is the augmentation ideal of $RN$, and each ideal $Φ(N)$ has a property : $RG/Φ(N)$ is $R$-algebra isomorphic to $R(G/N)$. Let $T$ be the set of non-unit ideals of $RG$. While the image of $Φ$ is rather a small subset of $T$, we give conditions on $R$ and $G$ for the image of $Φ$ to have some distribution in $T$. In the last section, we give criteria for choosing an element $x$ of $RG$ satisfying $RG/xRG$ is $R$-algebra isomorphic to $R(G/N)$ for a subgroup $N$ of $G$.
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