Exercise 22, § 3

a) Let $k$ be an ordered field, $A$ the quotient of the polynomial ring $k[X, Y, Z]$ by the principal ideal $(X^2 + Y^2 + Z^2 - 1)$; $x, y, z$ denote the canonical images of $X, Y, Z$ in $A$. Now prove that $A$ is factorial (consider the ring of fractions $A_{z-1}$ (notation of Chapter II, § 5, No. 1), and use No. 4, Prop. 3).

b) Let $(e_i)_{1 \leq i \leq 3}$ be the canonical basis of $A^3$, and let $M$ be the quotient of $A^3$ by the monogenic a-submodule $N$ generated by $xe_1 + ye_2 + ze_3$; now prove that $M$ is a projective a-module (form a supplementary submodule of $N$ in $A^3$).

c) Now prove that if $k = \mathbf{R}$, the a-module $M$ is not free (identify $M$ with a submodule of the module of continuous sections of the fiber space of tangent vectors to the unit sphere, and use the fact that there exists no continuous field of tangent vectors $\neq 0$ at every point of the sphere). \