I'm going through a course on number theory and there is a chapter on modules, in particular product of modules which I don't get. So somewhere it says :
"Let $I$ be an set and $(M_i)_{i\in I}$ a family of $R$-modules (and $R$ a ring)."
By my understanding $(M_i)_{i\in I}$ is just a set of $R$-modules. If for instance $I$ = $\{1,2,..,n\}$ , would $(M_i)_{i\in I}$ be $(M_1,M_2,..,M_n)$ ? Should I interpret an element, $x$ $\in$ $\prod_{i\in I}M_i$ as an $n$-tuple : $(x_i)_{i\in I}$ with $x_1 \in M_1$, $x_2 \in M_2$ and so on ?
Thanks for reading my question.
$(M_i)_{i\in I}$ is not $(M_1,M_2,\dots,M_n)$ (where $I=\{1,\dots,n\}$), because $(M_1,\dots,M_n)$ is the cartesian product of the family $\mathscr M:=(M_i)_{i\in I}$.
$\mathscr M$ is an indexed family of modules, that is for every $j\in I$ there is a corresponding $M_j\in\mathscr M$. In this way you have assigned every module in the given family $\mathscr M$ an index, i.e. a number by which you refer to this very module. The set $\{M_1,\dots,M_n\}$ (for finite $I$) has no such indexing per se (of course, you can construct one immediately, but think about infinite sets).
More formally, a family of sets $\mathscr S=(S_i)_{i\in I}$ is a function $f\colon I\to \mathscr S,~i\mapsto S_i$, that is every $i\in I$ gets assigned a $S_i\in\mathscr S$. In particular, there is no necessity for $I$ being finite, in contrast to your given example.
Back to your example. You are given a family of modules which are the building blocks for your product. But, incidentally, an element of the product $\prod_{i\in I} M_i$ (which then can be thought of as $(M_1,M_2,\dots)$) can be viewed of as $x=(m_i)_{i\in I}$ where $m_i\in M_i$. But the you have already added extra structure on the given family $\mathscr M$.