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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]
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]