Intermediate Rings that are not finitely generated [duplicate]

Let $R_1$ and $R_2$ are two rings such that $R_1$ $\subseteq$ $R_2$ and $R_2$ is a finitely generated ring over $R_1$. We know that, if $R_1$ is Noetherian and $S$ is a ring such that $R_1$ $\subseteq$ $S$ $\subseteq$ $R_2$ and $R_2$ is a finitely generated module over $S$, then $S$ is a finitely generated ring over $R_1$.

I found out in the June edition of The American Mathematical monthly that if we remove the condition that "$R_2$ is a finitely generated module over $S$", then there exists examples where $S$ might not be a finitely generated ring over $R_1$ and they construct an example using heavy machineries, I was thinking whether a 'simpler' example could be constructed.

And Also if one could provide a general algorithm on constructing such rings, it will be very much appreciable.

marked as duplicate by user26857 abstract-algebra StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Jun 21 '17 at 6:38

• @EricWofsey it means that my question has 2 parts, first part asks to give just an example of the fact and the second one asks to give a $general$ mechanism to construct such examples. – reflexive Jun 21 '17 at 4:30
• Tip: wrapping a text with asterisks produces italics. Using double asterisks gives bold. The spacing in such italics is better than what you get with $math$ $italics$, where letters are treated as symbols (as they should). If you prefer the latter, then I apologize. We can roll my edit back. – Jyrki Lahtonen Jun 21 '17 at 6:00
Let $R_1 = k, R_2 = k[x,y], S = k[x,xy,xy^2,xy^3,\dots]$.