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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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That $R/(a) \otimes R/(b) = R/(gcd(a,b))$? – Rankeya Dec 3 '12 at 21:59
You have asked $6$ questions on $MSE$ so far. You should accept answers to the previous questions you are satisfied with. – Rankeya Dec 3 '12 at 22:03
@ DonAntonio and Rankeya: I have now accepted the answers. I am new to this site and still learning how it works. Thanks for letting me know about this. – user49097 Dec 3 '12 at 23:17
@Rankeya: Yes, that is the theorem I am thinking of using, thanks. – user49097 Dec 3 '12 at 23:34
up vote 1 down vote accepted

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 $\,x,y\in R\,$:





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