The splitting length of mixed Abelian groups of rank one

summary:Suppose $G$ is a $p$-mixed splitting abelian group and $R$ is a commutative unitary ring of zero characteristic such that the prime number $p$ satisfies $p\notin \mathop {\text{inv}}(R) \cup \mathop {\text{zd}}(R)$. Then $R(H)$ and $R(G)$ are canonically isomorphic $R$-group algebras for any group $H$ precisely when $H$ and $G$ are isomorphic groups. This statement strengthens results due to W. May published in J. Algebra (1976) and to W. Ullery published in Commun. Algebra (1986), Rocky Mt. J. Math. (1992) and Comment. Math. Univ. Carol. (1995)