= First Tor of a cyclic quotient
{title2=$\operatorname{Tor}_1^A(A/I,M)=\ker(I\otimes_AM\to M)$}
Let $A$ be a <ring>, $I$ a <right ideal>, and $M$ a left <module>. Then $A/I$ and the resolving modules in this argument are right $A$-<modules>, while the second tensor factor is a left $A$-<module>. 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 $\operatorname{coker}(P_2\otimes_AM\to P_1\otimes_AM)$ with $I\otimes_AM$. Taking degree-one homology then gives the displayed kernel formula. Hence vanishing for every ideal is equivalent to the <flatness criterion using ideals>.
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