I heard that a field is always Noetherian and here Noetherian means that every ideal is finitely generated. Then, because a field has two ideals, 0 and the field itself, does this mean every field have to be finitely generated? Where I got it wrong?
2026-05-05 17:43:00.1778002980
A field is Noetherian
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"Finitely generated" means finitely generated as an ideal. An ideal $I\subseteq R$ is finitely generated if there exist finitely many elements $x_1,x_2,\ldots,x_n\in I$ such that for every $y\in I$ there exist $r_1,r_2,\ldots,r_n\in R$ such that $$y=r_1x_1+r_2x_2+\cdots+r_nx_n$$ In particular, for any unital ring the set $\{1 \}$ qualifies as a generating set for the ideal consisting of the entire ring.