Theorem 1

Let A be a commutative ring, $\mathfrak{g}$ an ideal of A and M an A-module. Consider the following properties: (i) M is afat A-module. (ii) $\mathrm{Tor}_1^A(N, M) = 0$ for every A-module N annihilated by $\mathfrak{z}$. (iii) $M / \mathfrak{g}M$ is ajlat $(A / \mathfrak{g})$-module and the canonical mapping $\mathfrak{z} \otimes_A M \to \mathfrak{g}M$ is bijective (the latter condition being equivalent to $\mathrm{Tor}_1^A(A / \mathfrak{g}, M) = 0$ by virtue of the relation $\mathrm{Tor}_1^A(A, M) = 0$ and the exact sequence

(iv) $M / \mathfrak{g}M$ is ajlat $(A / \mathfrak{g})$-module and the canonical homomorphism is bijective (property (GR) of § 2, no. 8). (v) For all $n \geq 1$, $M / \mathfrak{g}^n M$ is a flat $(A / \mathfrak{g}^n)$-module. Then (i) $\Rightarrow$ (ii) $\Leftrightarrow$ (iii) $\Rightarrow$ (iv) $\Leftrightarrow$ (v). If further $\mathfrak{g}$ is nilpotent or if A is Noetherian and M is ideally Hausdorff, properties (i), (ii), (iii), (iv) and (v) are equivalent.