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A question about submodules

  1. Jan 14, 2013 #1
    My textbook says that...

    If M is a left R-module, then a submodule N of M...is an additive subgoup N of M closed under scalar multiplication: [tex]rn \in N[/tex] whenever [tex]n \in N[/tex] and [tex]r \in R[/tex].

    So if we want to prove that something is a submodule, we need to show that...

    1) It closed under scalar multiplication
    2) The additive idenitity is in N
    3) N is closed under additition
    4) If x is in N, then so is its inverse

    Right?

    But, in the link that I attached, it only shows 1) and 3), right? Can anybody tell me why? Is the proof still considered complete?

    Thanks in advance
     

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  3. Jan 14, 2013 #2

    CompuChip

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    Isn't it as simple as: If (1) holds, set r = 0 to get (2) and r = -1 to get (4)?
    (OK, you might want to show that 0n is the additive identity for any n and that -n is the additive inverse of any n).
     
  4. Jan 14, 2013 #3
    Thanks.
     
  5. Jan 15, 2013 #4

    mathwonk

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    1 does not imply 2, unless the subset considered is non empty. i.e. 1 implies that IF the subset contains anything, then it also contains 0.
     
    Last edited: Jan 15, 2013
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