I've trouble with understanding some details about noetherian rings.
For example, I don't understand why this theorem is wrong:
A ring $R$ is noetherian if and only if it's finitely generated $\mathbb Z$-algebra.
My proof: the implication "noetherian $\Rightarrow$ finitely generated" is standard and obvious. The inverse implication also looks rather simple to me: cannot we say that finitely generated $\mathbb Z$-algebra is quotient $\mathbb Z[x_1, \ldots, x_n]$ / $\mathfrak I$ ($\mathfrak I$ is ideal), so by Hilbert basis theorem it's a quotient of the noetherian ring, so it's noetherian itself?
Can somebody help me and explain to me where is my mistake?
Best Answer
You should rethink the implication "noetherian $\Rightarrow$ finitely generated". Noetherianness only implies that every ideal is finitely generated as an ideal, but not necessarily as a ring. Every field is noetherian, for example, but an uncountable field cannot be finitely generated. In fact, no infinite field is finitely generated as a ring ; see If a ring is Noetherian, then every subring is finitely generated? for details.