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Let $B=k[X,Y]/(Y)$ be a module over $A=k[X,Y]/(XY)$ (under the natural ring-homomorphism). Give an example of torsion-free module $M$ over $A$ such that $M \otimes_{A} B$ is not torsion-free over $B$.

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You can take, for example, $M=A/(x)$, where $x$ is the residue class of $X$ modulo $(XY)$.

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