Freely generated module

Question

A very basic question but given an $R$-module $M$, if $S \subseteq M$ generates $M$ freely, must $S$ necessarily be finite or can it be infinite?

Answer 1

No, it can be infinite. If you like, by definition an $R$-module $M$ is freely generated by a subset $S$ exactly when when the map $\bigoplus\limits_{s \in S}R \to M$ defined by $1_s \mapsto s$ for any $s \in S$ is an isomorphism. So e.g. if $R$ is any nonzero ring and $M = \bigoplus\limits_{i \in \mathbb{Z}} R$ (a direct sum) then $M$ is freely generated by an infinite (in this case countable) subset.

Though be careful: famously e.g. the direct product $\prod\limits_{i \in \mathbb{Z}} \mathbb{Z}$ is not a free $\mathbb{Z}$-module, see here.