It is classical result of Adams that every $H\mathbb{Z}$-module spectra splits as a wedge of Eilenberg-MacLane spectra. Let me briefly recall what he writes about the proof.

Let $M$ be an $H\mathbb{Z}$-module spectrum. Adams constructs a map $$\alpha:\bigvee_k\Sigma^k S(\pi_kM)\rightarrow M$$ by taking the wedge of the maps $\Sigma^kS(\pi_kM)\rightarrow M$ inducing an isomorphism on $\pi_k$, where $SA$ denote the Moore spectrum on the abelian group $A$.

The map $\alpha$ induces a map of $H\mathbb{Z}$ by taking $\tilde{\alpha} = \mu \circ (1\wedge \alpha)$.

Now, $\tilde{\alpha}$ is without doubt a map of $H\mathbb{Z}$-modules, but why is it a weak equivalence?