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 $c=\mathrm{lcm(a,b)}$, $d=\gcd(a,b)$.
Modules over PID
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abstract-algebra
modules
principal-ideal-domains