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?


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.

  • $\begingroup$ 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

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