The quotient of a product is module isomorphic to the product of quotients

Let R be a ring. Let \{A_i\}_{i=1}^n be a finite family of left R-modules, and for each A_i let B_i \subseteq A_i be an R-submodule. Prove that (\prod A_i)/(\prod B_i) \cong_R \prod A_i/B_i.

In this previous exercise, we showed that the mapping \Phi : \prod A_i/\prod B_i \cong_R \prod A_i/B_i given by \Phi((a_i) + \prod B_i) = (a_i + B_i) is a well-defined group homomorphism. It suffices to show that \Phi preserves scalar multiplication. To that end, let r \in R. Then \Phi(r \cdot ((a_i) + \prod B_i)) = \Phi(r \cdot (a_i) + \prod B_i) = \Phi((r \cdot a_i) + \prod B_i) = (r \cdot a_i + B_i) = r \cdot (a_i + B_i) = r \cdot \Phi((a_i) + \prod B_i) as desired.

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