4
$\begingroup$

Suppose that $M,N$ are $R$-modules, $I,J\lhd R$. Suppose that $MI=JN=0$ and that $I+J=R$. Prove that $M\bigotimes_R N=0$.

This is very easy to prove if the ring is unital as you may write $1=i+j$, and then $m\otimes n=m1\otimes n=mi\otimes jn=0$. But what about the case when $R$ is not unital, is it still true then?

$\endgroup$
2
+100
$\begingroup$

If we do not assume that $R$ is unital, then we may consider the case where $R$ is a nonzero ring with zero multiplication and $M=N=I=J=R$. We get that $M\otimes_R N = M\otimes_{\mathbb Z} N$. This does not have to be zero, e.g. when $R$ is the group $\mathbb Z$ equipped with zero multiplication and $M=N=I=J=R$, then we get $M\otimes_{\mathbb Z} N \cong \mathbb Z$ as an additive group.

$\endgroup$
  • $\begingroup$ Thanks. This is what I figured as well, but it's from a comp exam so I wanted to check my sanity. $\endgroup$ – RougeSegwayUser Sep 24 '16 at 17:10

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.