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).