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 obviously Noetherian, but the subring $k[x_1,x_2,\dots]$ is not since there is an infinite ascending chain of ideals which never stabilizes.

Is there an instance of a

finitely generatedNoetherian ring over some ground ring $R$, that has an intermediate ring which is not finitely generated over $R$, and hence not Noetherian either?