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Let $a$ and $b$ be two nonzero elements of a PID $R$. Prove that direct sum $R/aR\oplus R/bR$ is isomorphic to the direct sum $R/cR\oplus R/dR$, where $c=\mathrm{lcm(a,b)}$, $d=\gcd(a,b)$.

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Hint: The structure theorem for modules over a PID gives a canonical form for finitely generated torsion modules. Here you have two finitely generated torsion modules, say $T_1$ and $T_2$, which you want to show are isomorphic, but neither one is (necessarily) in canonical form. So the natural first thing to do is...