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Let $B$, $C$, and $D$ be abelian groups such that $\mathbb Z\times B$ is isomorphic to $C\times D$. Is there a group $E$ such that $C$ or $D$ is isomorphic to $\mathbb Z\times E$?

Reference: page 263 of McKenzie, McNulty, and Taylor, Algebras, Lattices, Varieties: Volume I (Wadsworth & Brooks/Cole, Monterey, California, 1987; AMS Chelsea reprint).

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  • $\begingroup$ @TobiasFritz Thank you for the reference, but I don't see how this is a special case of Theorem 7 of Walker's paper, as we are not "cancelling" anything: all four groups could be different. $\endgroup$
    – Tri
    Commented Jul 20 at 3:38

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If I understand correctly, the question is equivalent to whether the class of abelian groups with no $\mathbf{Z}$ direct factor is stable under taking direct products.

But this is clearly true, since an abelian group $G$ has no $\mathbf{Z}$ direct factor iff $\mathrm{Hom}(G,\mathbf{Z})=0$.

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  • $\begingroup$ My comment thanking you for your answer seems to have disappeared. I hope moderators don't delete this one. $\endgroup$
    – Tri
    Commented Jul 20 at 5:31

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