Abelian $p$-groups $A$ and $B$ such that $\textrm {Tor}(A, G)\cong \textrm {Tor}(B, G),$ $G$ reduced

Abstract:

Let $A$ be an abelian $p$-group having all of its finite Ulm invariants nonzero. Let $C$ be a countable direct sum of cyclic $p$-groups such that for each nonnegative integer $n$, the $n$th Ulm invariant of $C$ is zero if the $n$th Ulm invariant of $A$ is finite. Then for all reduced abelian groups $G$, ${\operatorname {Tor}}(G,A) \cong {\text {Tor}}(G,A \oplus C)$.