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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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Since you are new, I want to give some advice about the site: To get the best possible answers, you should explain what your thoughts on the problem are so far. That way, people won't tell you things you already know, and they can write answers at an appropriate level; also, people are much more willing to help you if you show that you've tried the problem yourself. If this is homework, please add the [homework] tag; people will still help, so don't worry. Also, many would consider your post rude because it is a command ("Give..."), not a request for help, so please consider rewriting it. –  Zev Chonoles Dec 21 '12 at 6:18
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Are the downvotes really just because of the imperative "Give..."? –  S123 Apr 2 '13 at 14:30
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@ZevChonoles, what does it mean to "show that you've tried the problem yourself" where a problem asks for an example, the example is easy, and the OP does not see how to construct one. In such a situation there may not be any information to provide except that nothing has worked, there is no idea of where to look, or other useless negatives. Sometimes the process of finding the example is not progressive, you have to "see" it somehow and are stuck otherwise. –  zyx Apr 3 '13 at 9:33
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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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