First Tor of a cyclic quotient (${\mathrm{Tor}}_1^A(A/I,M)=\ker (I{\otimes }_{A}M\to M)$)

Start a free resolution of $A/I$ with $P_0=A$ and a free module $P_1$ surjecting onto $I$. The next differential presents the relations in $I$. Right exactness of the tensor product of modules identifies $\mathrm{coker}(P_2{\otimes }_{A}M\to P_1{\otimes }_{A}M)$ with $I{\otimes }_{A}M$. Taking degree-one homology then gives the displayed kernel formula. Hence vanishing for every ideal is equivalent to the flatness criterion using ideals.