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

## TL;DR

This paper explores the relationship between subgroups of an abelian group and certain ideals in its group ring, providing conditions for when these ideals produce specific quotient structures.

## Contribution

It characterizes when ideals associated with subgroups of an abelian group generate quotients isomorphic to group rings of quotient groups, extending known injections.

## Key findings

- Conditions for the distribution of subgroup-related ideals in the set of non-unit ideals.
- Criteria for selecting elements in the group ring to produce desired quotient isomorphisms.

## 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 $G$.