Exercise 9, § 2

Let $\Sigma$ be the species of division ring structures. Show that a notion of $\sigma$-morphisms for this species of structures may be defined by taking the morphisms of a division ring K into a division ring K$'$ to be the homomorphisms $f$ of K into K$'$ in the sense of Example 2 of no. 4, together with the mapping $f_0$ of K into K$'$ for which $f_0(0) = 0$ and $f_0(x) = 1$ for all $x \neq 0$ in K. Show that, with this notion of morphisms, we have the following properties : for every division ring K (of any characteristic) the division ring structure of K induces a division ring structure (isomorphic to that of $\mathbf{F}_2$) on the set $\{0, 1\}$; and that if R is the equivalence relation whose equivalence classes are $\{0\}$ and $\mathrm{K}^* = \mathrm{K} - \{0\}$, there exists a quotient structure (isomorphic to that of $\mathbf{F}_2$) of the structure of K by the relation R. \*