Is $\mathbb{Z}[\frac{1}{2}]$ Noetherian?

Question

I am a bit confused with the question wether $\mathbb{Z}[\frac{1}{2}]$ is a Noetherian ring or not.

I would say that $\mathbb{Z}[\frac{1}{2}]$ is a Noetherian ring.

Reason : $\mathbb{Z}[\frac{1}{2}]$ is a finitely generated $\mathbb{Z}$-module. Hence, it is Noetherian as a $\mathbb{Z}$-module. Since every ideal of $\mathbb{Z}[\frac{1}{2}]$ is also a $\mathbb{Z}$-module, it is a Noetherian ring.

Is this correct ?

Thanks for your help.

Answer 1

$\Bbb Z[\frac12]$ is not a finitely generated $\Bbb Z$-module.

But $\Bbb Z[\frac12]$ is a finitely generated $\Bbb Z$-algebra. It is isomorphic to a quotient of $\Bbb Z[X]$ which is Noetherian (by the Hilbert basis theorem) and so is itself Noetherian.

Answer 2

For $\alpha\in\mathbb C$, $\mathbb{Z}[\alpha]$ is a finitely generated $\mathbb{Z}$-module iff $\alpha$ is integral over $\mathbb Z$, which $1/2$ is not.

Answer 3

Note that $\mathbb{Z}[\frac{1}{2}]$ is not a finitely generated $\mathbb{Z}$-module, but finitely generated as $\mathbb{Z}$-algebra.