Let $A, B$ be $R$-modules. If $A \subseteq B$, then is $M \otimes A \subseteq M \otimes B$? What happens if $A$ is given to be a flat module?
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1@Lelouch: tensor product is right exact. It's left exact iff $M$ is flat. – Qiaochu Yuan Nov 23 '20 at 23:08
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@Geet: for fixed $M$, this is true for all $A \subseteq B$ iff $M$ is flat. – Qiaochu Yuan Nov 23 '20 at 23:09
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It is also true if $B/A$ is flat, for any $M$. – Bernard Nov 23 '20 at 23:12
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@QiaochuYuan Thanks, sorry. But what is a counterexample when $M$ is not flat, for example take $M = \mathbb{Z}_2, R = \mathbb{Z}$? – Lelouch Nov 24 '20 at 04:04
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@Lelouch: take $B = \mathbb{Z}, A = 2 \mathbb{Z}$. – Qiaochu Yuan Nov 24 '20 at 05:25