I have a lot of confusion in polynomial ring. Please help me explain it. Firstly, as we know, if $R$ is a ring then $R$ is sub-ring of $R[X]$ and $R[X]$ is a sub-ring of $R[[X]]$. I just wanna ask that how about the ring $R[X_{1},X_{2},...]$ which is a sub-ring with the same operations of $R[[X_{1},X_{2},...]]$. Is it Noetherian and can you find an ideal which is non-finitely generated in it? I know that is easy like $(X_{1}-\alpha_{1},X_{2}-\alpha_{2},....)$. I do not know is it ok? Moreover, denote it by $R_{\infty}$ then $R_{\infty}$ has a field of fractions, $L$ say. So why $L$ is Noetherian?
I see that if $D$ is a domain and $F$ is a field of fractions of it then $F$ is not $D$-module Noetherian. So is it contraction?