$\{A_i\}$ left $R$-modules and $\{B_i\}$ be submodules of $A_i$ for each $i$. Prove that:

$\frac{A_1 \oplus .... \oplus A_k}{B_1 \oplus .... \oplus B_k} \cong \frac{A_1}{B_1} \oplus \dots \oplus \frac{A_k}{B_k}$

Soooo use the natural projection mapping and show that the kernel equals $B_1 \oplus .... \oplus B_k$? And that it's an epimorphism? Can somebody help me with the details? What are the main differences to this as to what proving it for rings and ideals? Nothing I suppose since Ideals are R-Modules... anyway yeah I just wanted to post this here because I figured one of you smarty pants would have something useful to say about it. Thanks!

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    $\begingroup$ What details are you wanting? You have the right idea, so start writing things down. Every step is essentially obvious. $\endgroup$ – David Hill Nov 8 '18 at 18:39
  • $\begingroup$ yeah okay thanks dawg just wanted to make sure I wans't missing something in moving from rings and ideals to the more general setting of modules I guess $\endgroup$ – Math is hard Nov 8 '18 at 18:45

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