Subgroups of an abelian group, related ideals of the group ring, and quotients by those ideals

Abstract

Let $RG$ be the group ring of an abelian group $G$ over a commutative ring $R$ with identity. An injection $\Phi$ from the subgroups of $G$ to the non-unit ideals of $RG$ is well-known. It is defined by $\Phi(N)=I(R,N)RG$ where $I(R,N)$ is the augmentation ideal of $RN$, and each ideal $\Phi(N)$ has a property : $RG/\Phi(N)$ is $R$-algebra isomorphic to $R(G/N)$. Let $T$ be the set of non-unit ideals of $RG$. While the image of $\Phi$ is rather a small subset of $T$, we give conditions on $R$ and $G$ for the image of $\Phi$ 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 $G$.