Free module finite generated with a submodule not finite generated. Anyone knowns an example of a R-module finite genereted with a submodule not finite generated?
I find the following example: Taking the set of function $f:[0,1]\rightarrow\mathbb{R}$ seen as module of it self. This is finite generated. If we take the subset of functions $f$ such that $f(x)=0$  for all $x\in[0,1]$ except a finite numbers of points then we will get a submodule wich is not finite generated. Why?
This is a different example of the others I saw. 
Thanks a lot!!!

Let $R=k[x_1,x_2,\ldots,x_n,\ldots ]$. Now $R$ as a module over itself is finitely generated, but clearly $I=(x_1,x_2,\ldots,x_n,\ldots )$ is not finitely generated.
