$R/(a) \oplus R/(b) \cong R/\gcd(a,b) \oplus R/\operatorname{lcm}(a,b) $
Proving that $\mathbb{Z}_m\oplus \mathbb{Z}_n \cong \mathbb{Z}_d\oplus \mathbb{Z}_l $ as groups, where $l=\mathrm{lcm}(m,n)$ and $d=\gcd(m,n)$