Noetherian Rings: Does R Need to Be Finitely Generated? - Peter

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So $B$ is a finite set of generators.In summary, Dummit and Foote define a Noetherian ring as a commutative ring with 1 where every ideal is finitely generated. In the context of fields, this means that every field is Noetherian, and the only ideals are the trivial ideal {0} and the field itself. This does not necessarily mean that every field is finitely generated, as the field itself can serve as a finite set of generators.
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Dummit and Foote in Chapter 9 - Polynomial Rings define Noetherian rings as follows:

Definition. A commutative ring R with 1 is called Noetherian if every ideal of R is finitely generated.

Question: Does this mean that R itself must be finitely generated since R is an ideal of R?

This question is important in the context of fields since D&F go on to say that every field is Noetherian. In the case of fields the only ideals are the trivial ideal {0} and the field itself. But this would mean every field is finitely generated which does not seem to be corect.

Can anyone clarify these issues for me?

Peter

[This has also been posted on MHF]
 
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Peter said:
Question: Does this mean that R itself must be finitely generated since R is an ideal of R?

Take into account that for every commutative and unitary ring $R$ (noetherian or not), we have $R=R\cdot 1=(1)$.

But this would mean every field is finitely generated which does not seem to be corect.

Why? In such a case, $R$ is an $R$-vector space and a basis is $B=\{1\}$.
 

FAQ: Noetherian Rings: Does R Need to Be Finitely Generated? - Peter

What is a Noetherian ring?

A Noetherian ring is a commutative ring in which every ideal can be finitely generated. This means that for any subset of elements in the ring, there exists a finite set of elements that can generate the same ideal.

What is the significance of Noetherian rings in mathematics?

Noetherian rings have a lot of important applications in algebraic geometry, algebraic number theory, and commutative algebra. They also have connections to other areas of mathematics such as topology and representation theory.

Does R need to be finitely generated for it to be a Noetherian ring?

Yes, for a ring to be considered a Noetherian ring, it must be finitely generated. However, this does not necessarily mean that every ideal in the ring must be finitely generated.

How are Noetherian rings related to Hilbert's basis theorem?

Hilbert's basis theorem states that every ideal in a polynomial ring over a field is finitely generated. This theorem can be extended to Noetherian rings, which also have the property that every ideal is finitely generated.

Are there any other important properties of Noetherian rings?

Yes, Noetherian rings also have the property that every prime ideal is finitely generated, and every ascending chain of ideals eventually stabilizes. This makes them useful in proving many theorems in algebraic geometry and algebraic number theory.

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