# Invertible elements in generalized fields

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).

-
+1 For this amazingly fantastically cool reference! – Dylan Wilson Jan 31 '12 at 9:15
I don't think the tag "universal algebra" is appropriate. – Andreas Thom Jan 31 '12 at 9:44
@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. – Martin Brandenburg Jan 31 '12 at 9:46
@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. – Martin Brandenburg Jan 31 '12 at 9:49