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

[Math Amateur]

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].

 

[johng1]

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. (Handshake) (Smile)

 

[blue_raver22]

I am not well-versed in abstract algebra and may not be the best person to provide a response to this content. However, I can try to explain the concept of Noetherian rings and the exercise in question to the best of my understanding.

A Noetherian ring is a commutative ring in which every ascending chain of ideals eventually stabilizes. In other words, there cannot be an infinite increasing chain of ideals in a Noetherian ring.

The exercise is asking you to show that the ring of continuous real valued functions on [0, 1] is not Noetherian by exhibiting an explicit infinite increasing chain of ideals. This means you need to find a sequence of ideals in this ring that keeps getting larger and larger without ever stabilizing.

To do this, you can consider the sequence of ideals I_n = {f in C([0,1]) | f(x) = 0 for all x in [0, 1/n]}. Here, C([0,1]) represents the ring of continuous real valued functions on [0, 1].

This sequence of ideals is infinite and increasing because I_1 is a proper subset of I_2, which is a proper subset of I_3, and so on. This sequence will never stabilize because for any natural number n, there exists a function in I_{n+1} that is not in I_n. Therefore, the ring of continuous real valued functions on [0, 1] is not Noetherian.

I hope this explanation helps. Please seek further assistance from a mathematician or an expert in abstract algebra if needed.