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$\mathbb{Z}$-homomorphism and $\mathbb{Z}_p$-homomorphism

$\newcommand{\cts}{\mathrm{cts}}$Thanks for your reading. Let $A,B$ be two $\mathbb{Z}_p$-modules, where $\mathbb{Z}_p$ is the $p$-adic integer ring. I have two questions. Is $\mathrm{Hom}_{\mathbb{Z}...
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