Proposition 15.50.12. The following types of rings are G-rings:

1. fields,

2. Noetherian complete local rings,

3. $\mathbf{Z}$,

4. Dedekind domains with fraction field of characteristic zero,

5. 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 15.50.6. The statement on finite type overrings is Proposition 15.50.10. $\square$

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