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

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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