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Math Help - Modules

  1. #1
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    Modules

    Let R be a commutative ring and let J be an Ideal in R. Recall That if M is an R-Module, then JM= { sum jimi } is a submodule of M. Prove that M/JM is an R/J module if we define scalar multiplication: (r+J)(m+JM) = rm + JM.

    Conclude that If JM ={0}, then M itself is an R/J module. IF J is maximal ideal in R, then M is a vetor space over R/J.
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  2. #2
    Member Haven's Avatar
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    Re: Modules

    If JM=\{0\}, then it is not hard to prove that  M/JM \CONG M. Now you can define what scalar multiplcation by elements of (R/J) to M meeans using this isomorphism.

    Recall That if J is a maximal ideal of R, then R/J is a field. If M is an S-module, where S is a field, then M is an S-vector space.
    Thanks from mash
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