I'm trying to find an example showing that subrings of noetherian rings are not necessarily noetherian. So I just searched the net and this site, and came upon this:
"A common example showing that a subring of a Noetherian ring is not necessarily Noetherian is to take a polynomial ring over a field $k$ in infinitely many indeterminates, $k[x_1,x_2,\dots]$. The quotient field is then Noetherian obviously, but the subring $k[x_1,x_2,\dots]$ is not since there is an infinite ascending chain of ideals which never stabilizes."
I haven't had the chance to read my textbook the past week so I'm a bit behind on my understanding of noetherian rings. I understand that they are rings in which every chain of its ideals stabilizes. I also understand that a polynomial ring $R[x_1, x_2, ...]$ of noetherian ring $R$ is also noetherian. So what I don't get in the example is when they say polynomial ring, do they mean a ring that has infinite indeterminates or it could just be any polynomial. Another question is what are quotient fields and why is it obviously noetherian?