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Noetherian Rings - Dummit and Foote - Chapter 15 - Exercise 2a

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,918
In Dummit and Foote Chapter 15 Exercise 2(a) on page 668 reads as follows:

Show that the following ring is not Noetherian by exhibiting an explicit infinite increasing chain of ideals:

- the ring of continuous real valued functions on [0, 1]


I would appreciate help on this exercise.

Peter

[This has also been posted on MHF]
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,725
In Dummit and Foote Chapter 15 Exercise 2(a) on page 668 reads as follows:

Show that the following ring is not Noetherian by exhibiting an explicit infinite increasing chain of ideals:

- the ring of continuous real valued functions on [0, 1]


I would appreciate help on this exercise.
You could take the n'th ideal to be the set of continuous functions on [0,1] that vanish on the interval [0,1/n].
 

johng

Well-known member
MHB Math Helper
Jan 25, 2013
236
Another solution. Let the nth ideal be the principle ideal generated by the function \(\displaystyle f_n(x)=x^{1/n}\).
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,725
Another solution. Let the nth ideal be the principal ideal generated by the function \(\displaystyle f_n(x)=x^{1/n}\).
That is the algebraist's solution, mine was the analyst's solution. (Handshake) (Smile)