MHB Noetherian Rings - Dummit and Foote - Chapter 15

[Math Amateur]

Dummit and Foote Exercise 1 on page 668 states the following:

"Prove the converse to Hilbert's Basis Theorem: if the polynomial ring R[x] is Noetherian then R is Noetherian"

Can someone please help me get started on this exercise.

Peter

[Note: This has also been posted on MHF]

 

[mathbalarka]

(This is an old question, so I'll be posting a full answer instead of just hints)

Note that there is a canonical homomorphism $R[X] \to R$ given by $X \mapsto 0$. As Noetherianity is preserved by factoring, if $R[X]$ is Noetherian, then so is $R[X]/(X)$; and the latter is, as per the homomorphism, isomorphic to $R$. $\blacksquare$