Example 7

Let $a \in A$ be such that the mapping $a_A : x \mapsto ax$ of $A$ into $A$ is injective (“$a$ is not a left divisor of 0”). If $E$ is a flat $A$-module, then the homothety $a_E$ is injective, since it is identified with $a_A \otimes 1 : A_d \otimes_A E \to A_d \otimes_A E$. In particular, if $A$ is integral, every flat $A$-module is without torsion. Conversely, if $A$ is principal, every torsion-free $A$-module is flat: indeed, if the $A$-module $E$ is without torsion, every finitely generated submodule of $E$ is free (VII, § 4, No. 4, Cor. 2 to Theorem 4), and $E$ is a filtered increasing union of flat submodules, hence is flat (Prop. 4 (ii)).