I've tried to show that
A product of $\mathbb{Z}$ may not be a direct sum of $\mathbb{Z}$.
Since they are isomorphic when finite product, I thought that the infinite case matters.
I stuck here:
I may regard a member in a product of $\mathbb{Z}$ as
a function $X \to \mathbb{Z}$.
To get a contradiction, I might consider 'big' $X$.
However, what if there is soooo big Y such that the direct sum of $\mathbb{Z}$ –$Y$ times– is isomorphic to the product of $\mathbb{Z}$??
$$\mathbb{Z}^\mathbb{Z}\not\cong \bigoplus_{z \in \mathbb{Z}}\mathbb{Z}$$
Reason:
$$|\mathbb{Z}^\mathbb{Z}|= \aleph_0^{\aleph_0} = 2^{\aleph_0} = |\mathbb{R}|$$
while $$\left|\bigoplus_{z \in \mathbb{Z}}\mathbb{Z}\right| = \aleph_0 = |\mathbb{N}|$$