I don't understand this definition. Can someone explain it to me and give some simple examples?
Let $R$ be a ring, $M$ a left $R$-module and $A \subset M$ some set. The least (in terms of inclusion) submodule of module $M$ containing $A$ (i.e the intersection of all submodules of module $M$ containing $A$) we call the submodule generated by $A$ and mark $\left<A\right>$. Every set $A$ with the property that $\left<A\right>=M$ we call a set of generators of module $M$. If
$$ A=\{a_1, \dots ,a_n\}, $$
then we denote
$$ \left<a_1, \dots ,a_n\right>=\left<A\right>. $$
We say that a module is finitely generated (cyclic) if there exists a finite (with one element) set of generators.
Are there any equivalent definitions?
Example:
Let $R=\mathbb{Z}$ and $M$ an abelian group, then $M$ is a $\mathbb{Z}-$module. So $M$ is finitely generated $\mathbb{Z}-$module (or abelian group) iff there is a $A\subset M$ with $|A|<\infty$ s.t $$M=\langle A\rangle=\langle x_1,...,x_n\rangle$$ which means that every $m\in M$ is $$m=r_1x_1+...+r_nx_n, \quad r_i\in \mathbb{Z}$$
Equivalent definition:
$M$ is finitely generated $R-$module if and only if there is a surjective $R$-linear map: $$f:R^n\to M$$ for some $n\in \mathbb{N}$