Exercise 16, § 1

Let A be a ring such that there exists an A-module M with a generating system of n elements but containing a free system of $n + 1$ elements.

(a) Show that there exists in $A_s^n$ a free system of $n + 1$ elements.

(b) Deduce that there exists in M an infinite free system (construct such a system by induction using (a)).

(c) Let C be a ring, E the free C-module $C_s^{(N)}$, $(e_n)$ its canonical basis and A the endomorphism ring of E. Let $u_1$ and $u_2$ denote the endomorphisms of E defined by the conditions $u_1(e_{2n}) = e_n, \quad u_1(e_{2n+1}) = 0, \quad u_2(e_{2n+1}) = e_n, \quad u_2(e_{2n}) = 0$ for all $n \geq 0$. Show that $u_1$ and $u_2$ form a basis of the A-module $A_s$ and deduce that in the A-module $A_s$ there exist infinite free systems.

(d) Let A be the tensor algebra of a vector space E of dimension $\geq 2$ over a commutative field (cf. III, § 5, no. 1). Show that in the A-module $A_s$ there exist infinite free systems (observe that two linearly independent vectors in E are linearly independent in $A_s$ and use (b)). On the other hand, for every integer $n > 0$, every basis of the A-module $A_s^n$ has $n$ elements (cf. § 7, no. 2, Proposition 3).