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Math Help - R-Modules

  1. #1
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    R-Modules

    Let M = S_1 coproduct S_2 .... coproduct S_n. be a direct sum of R-modules. If T_i is a subset of S_i for all i prove that

    (S_1 coproduct .... coproduct S_n)/(T_1 coproduct ....T_n) is isomorphic to

    (S_1/T_1) co product (S_2/T_2) ---- coproduct (S_n/T_n).
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: R-Modules

    Quote Originally Posted by jcir2826 View Post
    Let M = S_1 coproduct S_2 .... coproduct S_n. be a direct sum of R-modules. If T_i is a subset of S_i for all i prove that

    (S_1 coproduct .... coproduct S_n)/(T_1 coproduct ....T_n) is isomorphic to

    (S_1/T_1) co product (S_2/T_2) ---- coproduct (S_n/T_n).
    Just do what's natural, define f:S_1\oplus\cdots\oplus S_n\to (S_1/T_1)\oplus\cdots\oplus(S_n/T_n) by (s_1,\cdots,s_n)\mapsto (s_1+T_1,\cdots,s_n+T_n). Is this map surjective? What is the kernel of this map? Why do you care?
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