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- Jun 22, 2012

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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]