$\def\Z{{\mathbb{Z}}\,} \def\Spec{{\rm Spec}\,}$ Suppose $R$ a ring and consider $\Spec(\prod_{i \in \mathbb{Z}} R)$. Now for the finite case, I know that holds $\Spec(R \times R) = \Spec(R) \coprod \Spec(R)$.
My intutition says that this does not extend to the infinite case. Maybe $\Spec(\oplus_{i \in \Z} R) = \coprod_{i \in \Z}\Spec(R)$ holds, but I am not sure. Can anybody give a proof or counter example for both the infinite direct sum and direct product?
The spectrum of an infinite direct product is complicated. For example, the spectrum of an infinite direct product $\prod_{i \in I} F_i$ of fields can canonically be identified with the space $\beta I$ of ultrafilters on $I$, also known as the Stone-Čech compactification of $I$. For more on this in the special case that each $F_i$ is $\mathbb{F}_2$ see this blog post.
In general the spectrum of an arbitrary infinite direct product $\prod_{i \in I} R_i$ fibers over $\beta I$, where the fiber over an ultrafilter $U \in \beta I$ is the spectrum of the ultraproduct $\prod_{i \in I} R_i / U$. For more on this see Eric Wofsey's excellent answer here and this blog post.