Let $M$ be an $R$-module (all rings are commutative). Let $S$ be a subring of $R$. Is there anyway to figure out what are all the possible $S$-module $N$ such that $N\otimes_{S}R\cong M$? In particular, what happened if $R^{\times}\bigcap S=S^{\times}$?
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The term you want to look up here is "faithfully flat descent." – Qiaochu Yuan Mar 16 '17 at 03:45
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@QiaochuYuan:thank you for giving me something to start. I had found this http://www.math.toronto.edu/~jacobt/Lecture9.pdf which is the only one I can understand, but it still seemed too hard. This is just a bonus question on the first midterm in a first undergraduate linear algebra course, I don't suppose it can be so complicated (don't worry I already submitted it). – caveat Mar 17 '17 at 05:53