The subring of $\mathbb{Q}[x]$ which contains all rational polynomials with constant term integer sits in between $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$ both of which is Noetherian, but this subring is not so although it has properties very much similar to UFD. Does this has any particular interest and any generalization?
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Yes, rings of the form $,R + x F[x],$ often prove handy for (counter)examples in ring theory. See the paper by Zafrullah cited here for a survey of their properties. – Bill Dubuque Mar 28 '24 at 07:16