TITLE: A module without a basis
QUESTION [12 upvotes]: It is well known that every vector space have a basis, after studying module theory I read in Wikipidia that not all modules have a basis.
Can someone please give an example of a module without a basis ?

REPLY [6 votes]: Another example, extracted from Musili's Introduction to Rings and Modules:

Any abelian group $M$ which has a nontrivial element of finite order
  cannot be free as a module over $\mathbb Z$.

In fact, say $B$ is a basis for $M$ over $\mathbb Z$. Let $a\in M\backslash\{0\}$ be such that $|a|=n$. Note that $n>1$. We have $$a=z_1b_1+z_2b_2+...+z_mb_m$$ for some $z_i\in \mathbb Z$ and $b_i\in B$, with $i=1,2,...,m$. Then, $$0=na=nz_1b_1+nz_2b_2+...+nz_mb_m.$$ By linearity of $B$, $$z_1=z_2=...=z_m=0 \Rightarrow a=0,$$ a contradiction. In particular,

Any finite abelian group is not free as a module over $\mathbb Z$.