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?

  • $\begingroup$ You have a few misconceptions here. A module is said to be cyclic if is has a generating set consisting of a single element. Think of cyclic groups as examples of cyclic $\mathbb{Z}$-modules. A module is said to be finitely generated if there exists a finite generating set of any size. So cyclic is a much stronger condition than finitely generated. An example of a non-finitely generated module is the Abelian group $\mathbb{Q}$ as this does not have a generating set of any finite size (can you prove this?) $\endgroup$ – Adam Higgins Mar 3 at 11:49
  • $\begingroup$ Which part don't you understand? $\endgroup$ – Carsten S Mar 3 at 12:18
  • $\begingroup$ Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. For equations, please use MathJax. $\endgroup$ – dantopa Mar 4 at 1:25


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 $n\in \mathbb{N}$

  • $\begingroup$ So even more simplifying, lets consider $M=Z_n$. We can write $\left<A\right>=\left<1,...,n-1\right>$ or even $\left<A\right>=\left<1\right>$ and both statements are correct? $\endgroup$ – Toidi Mar 3 at 12:52
  • $\begingroup$ Yes! For more accuracy you may use $[1],...,[n-1]$ instead of $1,...,n-1$ $\endgroup$ – giannispapav Mar 3 at 12:54

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