Let $(M_α)$ be a collection of $R$-modules, and $I$ is an ideal of the commutative ring $R$. Show that $(⊕M_α)/I(⊕M_α)$ is isomorphic to $⊕(M_α/IM_α)$ as $R/I$-modules.
Please help, thanks a lot!
2026-03-29 18:33:28.1774809208
Show $(⊕M_α)/I(⊕M_α)\simeq ⊕(M_α/IM_α)$ for ${M_α}$ $R$-modules, and $I$ is an ideal of the commutative ring $R$
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without tensor products
The obvious $R-$homomorphism $$ f:\oplus M_\alpha \rightarrow \oplus(M_\alpha/IM_\alpha) $$ is surjective and clearly satisfies $$ I(\oplus M_\alpha) \subseteq \operatorname{ker}(f) $$ so you have to prove the other inclusion, a fun exercise (not trivial!)
Once you have this, it is easy to show that the resulting $R-$isomorphism is $R/I$-linear.