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

#### Peter

##### Well-known member
MHB Site Helper
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
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
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
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.