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Durov's theory of generalized rings also includes generalized fields (5.7.6), which are defined as generalized rings, which are not subtrivial and whose proper strict quotients are subtrivial. For example, a classical ring is a generalized field iff it is a classical field. Nonclassical examples include $F_1$ ("the field with one element)" and element"), $F_{\infty}$ ("the residue field of the valuation ring of $\mathbb{R}$").\mathbb{R}$") and $F_{\emptyset}$ (the initial field).

Now Durov mentions after 6.1.16 (6.1.16) that every generalized field $K$ embeds into its generalized ring of total fractions $K':=T^{-1} K$ and that $K'$ is a generalized field such that every non-zero element of $|K'|$ is invertible.

Question 1. Is there a generalized field $K$ such that $K \neq K'$. In , or in other words, is there a generaloiized field such that not every non-zero element of $|K|$ is invertible?

Durov mentions in 5.7.9 (5.7.9) that this is unclear. I think that there should be a counterexample, but not within the generalized fields mentioned above.

Question 2. Does every non-subtrivial generalized domain $A$ embed into a generalized field?

Note that $A$ embeds into its total ring of fractions $A'$, which has again the property that every non-zero element of $|A'|$ is invertible, but it is not true in general that $A'$ is a generalized field ($A=\mathbb{F}_{\emptyset}[T]$ is easily seen to be a counterexample, here $|A| = (\mathbb{N},+)$ \{T^n : n \in \mathbb{N}\}$ and $|T^{-1} A| = (\mathbb{Z},+)$ \{T^z : z \in \mathbb{Z}\}$ has lots of proper quotients).
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Durov's theory of generalized rings also includes generalized fields (5.7.6), which are defined as generalized rings, which are not subtrivial and whose proper strict quotients are subtrivial. For example, a classical ring is a generalized field iff it is a classical field. Nonclassical examples include $F_1$ ("the field with one element)" and $F_{\infty}$ ("the residue field of the valuation ring of $\mathbb{R}$").

Now Durov mentions after 6.1.16 that every generalized field $K$ embeds into its generalized ring of total fractions $K':=T^{-1} K$ and that $K'$ is a generalized field such that every non-zero element of $|K'|$ is invertible.

Question 1. Is there a generalized field $K$ such that $K \neq K'$, i.e. In other words, is there a generaloiized field such that not every non-zero element of $|K|$ is invertible?

Durov mentions in 5.7.9 that this is unclear. I think that there should be a counterexample, but not within the generalized fields mentioned above.

Question 2. Does every non-subtrivial generalized domain $A$ embed into a generalized field?

Note that $A$ embeds into its total ring of fractions $A'$, which has again the property that every non-zero element of $|A'|$ is invertible, but it is not true in general that $A'$ is a generalized field ($A=\mathbb{F}_{\emptyset}[T]$ is easily seen to be a counterexample, here $|A| = (\mathbb{N},+)$ and $|T^{-1} A| = (\mathbb{Z},+)$ has lots of proper quotients).
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Durov's theory of generalized rings also includes generalized fields (5.7.6), which are defined as generalized rings, which are not subtrivial and whose proper strict quotients are subtrivial. For example, a classical ring is a generalized field iff it is a classical field. Nonclassical examples include $F_1$ ("the field with one element)" and $F_{\infty}$ ("the residue field of the valuation ring of $\mathbb{R}$").

Now Durov mentions after 6.1.16 that every generalized field $K$ embeds into its generalized ring of total fractions $K':=T^{-1} K$ and that $K'$ is a generalized field such that every non-zero element of $|K'|$ is invertible.

Question 1. Is there a generalized field $K$ such that $K \neq K'$, i.e. that not every non-zero element of $|K|$ is invertible?

I think that there should be a counterexample, but not within the fields mentioned above.

Question 2. Does every non-subtrivial generalized domain $A$ embed into a generalized field?

Note that $A$ embeds into its total ring of fractions $A'$, which has again the property that every non-zero element of $|A'|$ is invertible, but it is unclear whether not true in general that $A'$ is a generalized field ($A=\mathbb{F}_{\emptyset}[T]$ is easily seen to be a counterexample, here $|A| = (\mathbb{N},+)$ and $|T^{-1} A| = (\mathbb{Z},+)$ has lots of proper quotients).
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