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Noetherian Rings - Dummit and Foote - Chapter 15

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,918
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

Well-known member
MHB Math Helper
Mar 22, 2013
573
(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 $R$. $\blacksquare$