Proposition 15.49.12. The following types of rings are G-rings:

fields,

Noetherian complete local rings,

$\mathbf{Z}$,

Dedekind domains with fraction field of characteristic zero,

finite type ring extensions of any of the above.

Proposition 15.49.12. The following types of rings are G-rings:

fields,

Noetherian complete local rings,

$\mathbf{Z}$,

Dedekind domains with fraction field of characteristic zero,

finite type ring extensions of any of the above.

**Proof.**
For fields, $\mathbf{Z}$ and Dedekind domains of characteristic zero this follows immediately from the definition and the fact that the completion of a discrete valuation ring is a discrete valuation ring. A Noetherian complete local ring is a G-ring by Proposition 15.49.6. The statement on finite type overrings is Proposition 15.49.10.
$\square$

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