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}\}$has lots of proper quotients). 3 added 106 characters in body Durov's theory of generalized rings also includes generalized fields (5.7.6), which are defined as generalized rings, which are not subtrivial and whose proper strict quotients are subtrivial. For example, a classical ring is a generalized field iff it is a classical field. Nonclassical examples include$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},+)$has lots of proper quotients). 2 added 177 characters in body Durov's theory of generalized rings also includes generalized fields (5.7.6), which are defined as generalized rings, which are not subtrivial and whose proper strict quotients are subtrivial. For example, a classical ring is a generalized field iff it is a classical field. Nonclassical examples include$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).