In my module theory course, we recently proved the following theorems:
For a fixed ring $R$ and index set $I$, let $P_i$ and $Q_i$ be $R$-modules for all $i\in I$. Then $$P=\bigoplus _{i\in I}P_i$$ is projective if and only if each $P_i$ is projective, and $$Q=\prod_{i\in I}Q_i$$ is injective if and only if each $Q_i$ is injective.
But what exactly is the product of modules in the injective case? In our course we showed that the direct sum $M\oplus N$ of two $R$-modules is both the product and coproduct (in the categorical sense), so clearly the injective case is not referring to coproducts.
I tried asking my instructor during class, wondering whether he was simply using different notation for the direct sum, but he only said that the second assertion is not true if one deals with direct sums of modules, so that rules out that theory as well.
My last idea was that it simply meant the Cartesian product in the set theoretical sense, but then what is the module structure?
So, what is the second part of the theorem referring to?
The definition of the product of modules in general is that $R$ is a ring, and $M_i$, $i\in I$ are $R$-modules, then the product $M=\prod_{i\in I}M_i$ is the $R$-module of tuples $\{ (m_i)\mid m_i\in M_i\}$ with componentwise addition and "diagonal" multiplication $r.(m_i)_{i\in I}$. The direct sum $\oplus_{i\in I}M_i$ of $R$-modules $M_i$ is defined as the submodule of the direct product consisting of all $(m_i)$ satisfying $m_i=0$ for almost all $i\in I$. In general, an infinite product is not isomorphic to an infinite sum. For example, the module $M=\prod_{i\ge 0}\mathbb{Z}$ is not free-abelian, but $N=\oplus_{i\ge 0}\mathbb{Z}$ is free-abelian.