Generated submodule, cyclic and finitely generated modules, module basis and free module

Let $M$ be a left $R$-module and $S\subseteq M$. The submodule generated by $S$ is

$\langle S\rangle_R:=\bigcap\{N\le M:S\subseteq N\}.$

The family is nonempty because $M\le M$, and its intersection is a submodule by The one-step submodule criterion; intersections and sums of submodules are submodules. Thus $\langle S\rangle_R$ is the smallest submodule of $M$ containing $S$, just as The subgroup $\langle S \rangle$ generated by a subset, the cyclic subgroup $\langle g \rangle$, and cyclic groups defines a generated subgroup.

The module $M$ is cyclic if $M=\langle m\rangle_R$ for some $m\in M$, and finitely generated if $M=\langle S\rangle_R$ for some finite $S$. A subset $B\subseteq M$ is a basis if every element of $M$ has a unique expression as a finite $R$-linear combination of elements of $B$. A module possessing a basis is a free $R$-module.