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I am working on the following problem:

Let $R$ be a PID and let $a,b \in R$ be such that $\gcd(a,b) = 1$. Prove that there are $s,t \in R$ such that $sa+tb = 1$, that the $R$-module $R/\langle a \rangle \oplus R/\langle b \rangle$ is isomorphic to the $R$-module $R/\langle ab \rangle$, and that $R$-module $R/\langle a \rangle \otimes R/\langle b \rangle$ is isomorphic to the trivial $R$-module $0$.

I am thinking to use a well-known theorem on tensor products involving the gcd but I don't recall what the theorem is. I would greatly appreciate any help with this. Thank you.

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    @Rankeya: Yes, that is the theorem I am thinking of using, thanks.2012-12-03

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So we have that $\,\exists \,s,t\in R\,\,\,s.t.\,\,\,sa+tb=1$ , and then putting $\,I_a:\langle a\rangle\,\,,\,I_b:=\langle b\rangle\,$ , we get for any $\,x,y\in R\,$:

$(x+I_a)\otimes(y+I_b)=xy(1+I_a)\otimes(1+I_b)=(xysa+xytb)(1+I_a)\otimes(1+I_b)=$

$=\left[xysa(1+I_a)\otimes(1+I_b)\right]+\left[xytb(1+I_a)\otimes(1+I_b)\right]=$

$xys\left[(a+I_a)\otimes(1+I_b)\right]+xyt\left[(1+I_a)\otimes(b+I_b)\right]=xys(0)\otimes(1+I_b)+xt(1+I_a)\otimes(0)=$

$=0+0=0$