Cancellation of direct sums of countable abelian $p$-groups

Let $B \oplus A_{1} = C \oplus A_{2}$ be abelian groups where $B \cong C$ is a direct sum of countable $p$-groups. A condition is given on the Ulm-Kaplansky $p$-invariants of $B, A_{1}$ and $A_{2}$ such that $A_{1} \cong A_{2}$.