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.