Remark

If $A / \mathfrak{g}$ is a field (as often happens in applications) the condition "M/ $\mathfrak{g}M$ is a flat $(A / \mathfrak{g})$-module" holds automatically for every A-module M, which simplifies the statement of properties (iii) and (iv); moreover, in this case, property (v) is equivalent to saying that $M / \mathfrak{g}^n M$ is a free $(A / \mathfrak{g}^n)$-module for every integer $n \geq 1$ (Chapter 11, § 3, no. 2, Corollary 2 to Proposition 5).