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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 thenon-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 $\Phi(N)$ hasa property : $RG/\Phi(N)$ is $R$-algebra isomorphic to $R(G/N)$. Let $T$ be theset of non-unit ideals of $RG$. While the image of $\Phi$ is rather a smallsubset of $T$, we give conditions on $R$ and $G$ for the image of $\Phi$ tohave some distribution in $T$. In the last section, we give criteria forchoosing an element $x$ of $RG$ satisfying $RG/xRG$ is $R$-algebra isomorphicto $R(G/N)$ for a subgroup $N$ of $G$.

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