  An $S$-ring (a Schur ring) is said to be separable with respect to a class of
groups $\mathcal{K}$ if every algebraic isomorphism from the $S$-ring in
question to an $S$-ring over a group from $\mathcal{K}$ is induced by a
combinatorial isomorphism. A finite group is said to be \emph{separable} with
respect to $\mathcal{K}$ if every $S$-ring over this group is separable with
respect to $\mathcal{K}$. We provide a complete classification of abelian
$p$-groups separable with respect to the class of abelian groups.
