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?

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.

• Thank you very much! I've already found a mistake in this implication... Jul 2, 2019 at 10:40
• Could you say me, please, is it true that a submodule of a finitely generated module over finitely generated $\mathbb Z$-algebra is finitely generated? Is it true for quotients? Jul 2, 2019 at 10:45
• It is easy to see for quotients, and for submodules, see this question. Jul 2, 2019 at 10:51
• Thank you for your attention and help. Jul 2, 2019 at 10:53