Exercise 20, § 1

(a) Let A be a ring admitting a field of left fractions (I, § 9, Exercise 15). Let E be an A-module; show that, if in E $(x_i)_{1 \leq i \leq r}$ is a free system and $y, z$ two elements such that $x_1, \ldots, x_r, y$ on the one hand and $x_1, \ldots, x_r, z$ on the other are two related systems, then $x_2, \ldots, x_r, y, z$ is a related system. (b) Let A be the quotient ring $\mathbf{Z}/6\mathbf{Z}$ and let $(e_1, e_2)$ be the canonical basis of $A^2$; show that if $a = 2e_1 + 3e_2$, $a$ and $e_1$ form a related system, as do $a$ and $e_2$, although $e_1$ and $e_2$ form a free system.