Asking about $R$-submodules which are not finitely generated. 
Let $R=\mathbb{Z}[X_{1},X_{2},...,X_{n},...]$ the polynomial ring in infinitely many variables with coefficients in $\mathbb{Z}$. Find (and explain why) a $R^{R}$-submodule (being $R^R$ the regular module over $R$) which is not finitely generated.

I need help about this issue, thank you so much.

This is, I believe, the standard example of the fact that a submodule of a finitely generated module needs not be finitely generated.
Hint: The regular module is finitely generated (as it always is for a unital ring) by $1$. We are looking for a submodule (an ideal of $R$, basically) which is not finitely generated. $R$ is a polynomial ring. What kinds of ideals do you have in a polynomial ring? Can you think of one that's not finitely generated?
