Geodesic
On group rings of abelian $p$-groups of any cardinality
Matematičeskie zametki, Tome 6 (1969) no. 4, pp. 381-392.

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The problem is studied of the connection between an Abelian $p$-group $G$ of arbitrary cardinality and its group ring $LG$, where $L$ is a ring with unity nonzero characteristic $n\equiv0(\mod p)$, with $p$ being a prime. In particular, it is shown that group ring $LG$ defines to within isomorphism the basis subgroup of group $G$. If reduced Abelian $p$-group $G$ has finite type and if its Ulm factors decompose into direct products of cyclic groups, then group ring $LG$ determines group $G$ to within isomorphism. 
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     author = {S. D. Berman and T. Zh. Mollov},
     title = {On group rings of abelian $p$-groups of any cardinality},
     journal = {Matemati\v{c}eskie zametki},
     pages = {381--392},
     publisher = {mathdoc},
     volume = {6},
     number = {4},
     year = {1969},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1969_6_4_a2/}
}
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S. D. Berman; T. Zh. Mollov. On group rings of abelian $p$-groups of any cardinality. Matematičeskie zametki, Tome 6 (1969) no. 4, pp. 381-392. http://geodesic.mathdoc.fr/item/MZM_1969_6_4_a2/