Exercise 22, § 3

(a) Let $k$ be an ordered field and $A$ the quotient of the polynomial ring $k[X, Y, Z]$ by the principal ideal $(X^2 + Y^2 + Z^2 - 1)$; let $x, y, z$ denote the canonical images of $X, Y, Z$ in $A$. Show that $A$ is a factorial domain (consider the ring of fractions $A_{z-1}$ (notation of Chapter 11, § 5, no. 1), using no. 4, Proposition 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 monogenous sub-A-module $N$ generated by $xe, + ye, + te,;$ show that $M$ is a projective $A$-module (form a complementary submodule of $N$ in $A^3$).

(c) Show 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 fibre bundle 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). \*