Theorem 2

Let E be a right $A$-module. The following conditions are equi- valent :

(i) E is flat ;

(ii) for every left $A$-module F, and every integer $n>0$, one has

(iii) For every cyclic left $A$-module $F$ of finite presentation, we have

(iv) For every left ideal $a$ of $A$ of finite type, the canonical mapping $E \otimes_A a \to E$ is injective; (v) For every exact sequence of right $A$-modules of the form and every left $A$-module $F$, the sequence is exact. (i) $\Rightarrow$ (ii): This is the corollary to Prop. 5 of X, p. 69. (ii) $\Rightarrow$ (iii): This is trivial. (iii) $\Leftrightarrow$ (iv): Every cyclic left $A$-module of finite presentation is isomorphic to a quotient $A/a$, where $a$ is a left ideal of finite type, so (iii) is equivalent to (iv) according to X, p. 72, Example. (iii) $\Rightarrow$ (i): According to X, p. 8, Prop. 3, X, p. 72, Th. 1, $E$ is flat as soon as $\operatorname{Tor}_1^A(E, F) = 0$ for every left $A$-module $F$. If (iii) is satisfied, this is the case when $F$ is cyclic and of finite presentation. According to X, p. 11, Prop. 7, every $A$-module (resp. every cyclic $A$-module) is a filtered inductive limit of modules of finite presentation (resp. of cyclic modules of finite presentation); hence, according to X, p. 70, Prop. 8, it suffices to prove that if $\operatorname{Tor}_1^A(E, F) = 0$ when $F$ is cyclic, then this is also the case when $F$ is of finite type. We therefore proceed by induction on the cardinal of a system of generators $(f_1, \ldots, f_n)$ of $F$; the exact sequence gives rise to an exact sequence so that $\operatorname{Tor}_1^A(E, F) = 0$ since $\operatorname{Tor}_1^A(E, Af_1) = 0$ and $\operatorname{Tor}_1^A(E, F/Af_1) = 0$ by the induction hypothesis. (i) $\Rightarrow$ (v): This is Corollary 2 to Th. 1 (X, p. 72). (v) $\Rightarrow$ (iii): The exact sequence (X, p. 50) gives rise, for every left $A$-module $F$, to an exact sequence If (v) is satisfied, we have $\operatorname{Tor}_1^A(E, F) = 0$, whence (iii).