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

[Peter]

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]

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]

Another solution. Let the nth ideal be the principle ideal generated by the function \(\displaystyle f_n(x)=x^{1/n}\).

 

[Opalg]

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.