Let $F$ be a field and let $R$ be the ring of polynomials over $F$ in infinitely many variables $x_1, x_2, \dots, x_n, \dots$. (Of course, each element of $R$, being a polynomial, will involve only finitely many of the $x_i$'s, but the number of such variables need not be bounded.) How can I show that $R$, viewed as a module over itself, contains a submodule which is not finitely generated? Also that $R$ itself is finitely generated as an $R$-module.
Modules are not my forte
Thanks
$R$ is obviously generated by $1$ as an $R$-module. The ideal (submodule) $(x_1,x_2,\ldots)$ of $R$ is not finitely generated. For otherwise $(x_1,x_2,\ldots)=(f_1,\ldots,f_k)$ for some $f_1,\ldots,f_k\in R$, and every $f_i$ involves only finitely many indeterminates.