# Show $\frac{A_1 \oplus … \oplus A_k}{B_1 \oplus … \oplus B_k} \cong \frac{A_1}{B_1} \oplus \dots \oplus \frac{A_k}{B_k}$

$$\{A_i\}$$ 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!

• What details are you wanting? You have the right idea, so start writing things down. Every step is essentially obvious. – David Hill Nov 8 '18 at 18:39
• 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 – Math is hard Nov 8 '18 at 18:45