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.