TITLE: A field is Noetherian
QUESTION [6 upvotes]: 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?

REPLY [7 votes]: "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.