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Consider an integer polynomial ring, $A = \mathbb{Z}[t]$, and a ring of fractions, $B = \mathbb{Z}[t, t^{-1}]$; obviously, $A$ is a subring of $B$.

Now we consider two modules over $A$ and $B$, $M$ and $N$. We want to construct a map from $N$ to $M$. But the question is that the two modules are not over the same polynomial ring. So how can we make it?

Great thanks!

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    $\begingroup$ Please see the FAQ mathoverflow.net/faq for a list of sites where your question would be probably more appropriate. $\endgroup$
    – user2035
    Commented Sep 17, 2011 at 10:00
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    $\begingroup$ Have you tried to apply the "change of ring" technique ? For it, one uses a ring homomorphism $\varphi: R \to S$ and considers a $S$-Module $N$ as $R$-module in the obvious way. If $M$ is a $R$-module, then look for $R$-module homomorphisms $M \to N$. In your example $\varphi$ can be taken to be the inclusion $A \hookrightarrow B$. $\endgroup$
    – Todd Leason
    Commented Sep 17, 2011 at 12:29
  • $\begingroup$ I doubt that there is any site where this question would be appropriate. $\endgroup$
    – Steven Landsburg
    Commented Sep 17, 2011 at 13:43

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I do not think you can map a module over a smaller ring into a module over a larger ring in this case.$ \phi(rx)=r\phi(x)$ will not satisfy.

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