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Let $R$ be a PID and consider the ideals $(a),(c),(ac) \subset R$. Consider R/(ac) and R/(a) x R/(c).

Is there a natural surjective homomorphism from one of these modules to the other?

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    $\begingroup$ Maps into a product are equivalent to pairs of maps into the factors. For modules, maps from a direct sum (equivalently, direct product when only finitely many factors) are equivalent to maps from each summand/factors. $\endgroup$ – Arturo Magidin Nov 23 at 1:33

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