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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"), $F_{\infty}$ ("the residue field of the valuation ring of $\mathbb{R}$") and $F_{\emptyset}$ (the initial field).

Now Durov mentions (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 $K \neq K'$, or in other words, such that not every non-zero element of $|K|$ is invertible?

Durov mentions (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 $|A| = \{T^n : n \in \mathbb{N}\}$ and $|T^{-1} A| = \{T^z : z \in \mathbb{Z}\}$ has lots of proper quotients).

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  • $\begingroup$ +1 For this amazingly fantastically cool reference! $\endgroup$
    – Dylan Wilson
    Commented Jan 31, 2012 at 9:15
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    $\begingroup$ I don't think the tag "universal algebra" is appropriate. $\endgroup$
    – Andreas Thom
    Commented Jan 31, 2012 at 9:44
  • $\begingroup$ @Dylon: Indeed, Durov's theory develops commutative algebra and algebraic geometry in an amazingly general framework (and not just for the sake of generality, the compactification of $\mathrm{Spec}(\mathbb{Z})$ is one of the main motivations) and unifies various theories which seemed to be independent from each other. Within the $F_1$-context it has already been cited many times, but generalized rings don't seem to be established yet. I hope that this will happen soon. $\endgroup$
    – Martin Brandenburg
    Commented Jan 31, 2012 at 9:46
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    $\begingroup$ @Andreas: On the one hand I agree that this is not classical universal algebra, on the other hand the category of generalized rings is a full subcategory of the category of algebraic monads, which is equivalent to the category of Lawvere theories - so we are in universal algebra. $\endgroup$
    – Martin Brandenburg
    Commented Jan 31, 2012 at 9:49

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