Exercise 23, § 2

(a) Let $0 \to R \to L \to E \to 0$ be an exact sequence of left A-modules, where L is a free A-module; let $(e_i)$ be a basis of L. Show that the following conditions are equivalent:

(a) E is flat.

(b) For all $x \in R$, if $a$ is the right ideal generated by the components of $x$ with respect to the basis $(e_\alpha)$, then $x \in Ra$.

(y) For all $x \in R$, there exists a homomorphism $u_x : L \to R$ such that $u_x(x) = x$.

(8) For every finite sequence $(x_i)_{1 \leq i \leq n}$ of elements of R, there exists a homomorphism $u : L \to R$ such that $u(x_i) = x_i$ for $1 \leq i \leq n$. (Use the Corollary to Proposition 7 of no. 6.)

(b) Let a be a left ideal of A such that $A/a$ is a flat A-module. Show that, for every finitely generated left ideal $b \subset a$, there exists $x \in A$ such that

(use condition (8) of (a)).

(c) Derive from (a) a new proof of the result of Exercise 15.

(d) Let $r$ be the Jacobson radical of $A$ and $0 \to R \to L \to E \to 0$ an exact sequence of left $A$-modules such that $L$ is free. Suppose that $E$ is flat and $R$ is contained in $rL$. Show that $R = 0$ (in the notation of (a) observe that $a_r$ is a finitely generated ideal and that $a_r = a_{x,r}$).

(e) Let $E$ be a finitely generated flat $A$-module; suppose that there exists a two-sided ideal $b$ of $A$ contained in the Jacobson radical of $A$ such that $E/bE$ is a free $(A/b)$-module. Show that $E$ is then a free $A$-module (observethat there exists a finitely generated free $A$-module $L$ such that $L/bL$ is isomorphic to $E/bE$ and use Algebra, Chapter VIII, § 6, no. 3, Corollary 4 to Proposition 6; then apply (d)).