I am reading Dummit and Foote Chapter 15, Section 15.1: Noetherian Rings and Affine Algebraic Sets.

Exercise 9 reads as follows:


For k a field show that any subring of a polynomial ring k[x] containing k is Noetherian.

Give an example to show that such subrings need not be UFDs.


Can someone please help me get started on this problem?