Let $f:L\to G$ be a homomorphism of modules over commutative ring $R$. Show that if $f_{M}: L_{M} \to G_{M}$ is surjective for every maximal ideal $ M$ of $R$ then $f$ is surjective.
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2This is Proposition 3.9 of Atiyah-Macdonald. – Alex Wertheim Jul 21 '16 at 02:47
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It is just injective. I try to do in surjective case. But I got stuck. – Thế Long Lê Jul 21 '16 at 03:00
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4Localization is exact, hence - by looking at the cokernel - you are reduced to show that a module is zero if and only if all localizations are zero. If this is not well known to you, figure out a proof yourself or use search. – MooS Jul 21 '16 at 04:57
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With your hints I got it :D. Thank you so much :). – Thế Long Lê Jul 21 '16 at 08:42