Is every module a direct limit of cyclic modules?

Question

I want to show that

$M$ is $A$-flat is equivalent to $\mathrm{Tor}_1^A(M,A/I)=0$ for every finitely generated ideal $I$.

I want to show $\mathrm{Tor}^A_1(M,N)=0$ for any $A$-module $N$.

Is every module a direct limit of $A/I$ described above? (If this holds, we can pass from $\mathrm{Tor}^A_1(M,A/I)=0$ to $\mathrm{Tor}^A_1(M,N)=0$ by taking direct limit.)

Answer 1

$\mathrm{Tor}_1^A(M,A/I)=0$ is equivalent to the following: the canonical homomorphism $I\otimes_AM\to M$ (that is, $a\otimes m\mapsto am$) is injective, and this is Theorem 7.7 from Matsumura, Commutative Ring Theory.