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I Submodules A + B and A intersect B ... Blyth Ch. 2

  1. Aug 15, 2016 #1
    I am reading T. S. Blyth's book "Module Theory: An Approach to Linear Algebra" ... ... and am currently focussed on Chapter 2: Submodules; Intersections and Sums ... ...

    I need help with understanding two claims that Blyth makes concerning submodules ...

    The relevant text is as follows: ( see end of post for other text that may be relevant)



    ?temp_hash=73c4113bcdd90e05481e0782497ccecb.png



    I have two questions concerning the above text ... ...



    Question 1


    In the above text we read:

    " ... ... We know that ##A + B## is the smallest submodule of ##M## that contains both ##A## and ##B##, ... ... "


    My question is: how exactly do we know this ... ? How would we formally and rigorously prove this ... ?



    Question 2


    In the above text we read:

    " ... ... and that ##A \cap B## is the largest submodule contained in both ##A## and ##B##, ... ... "


    My question is: how exactly do we know this ... ? How would we formally and rigorously prove this ... ?


    Hope that someone can help with the above two questions ...

    Peter

    ==============================================================================

    PS Just in case readers need to reference some of Blyth's definitions or theorems in Chapter 2, I am providing the relevant text as follows:


    ?temp_hash=73c4113bcdd90e05481e0782497ccecb.png
    ?temp_hash=73c4113bcdd90e05481e0782497ccecb.png
    ?temp_hash=73c4113bcdd90e05481e0782497ccecb.png
    ?temp_hash=73c4113bcdd90e05481e0782497ccecb.png
    ?temp_hash=73c4113bcdd90e05481e0782497ccecb.png
     

    Attached Files:

  2. jcsd
  3. Aug 16, 2016 #2

    fresh_42

    Staff: Mentor

    How are ##A+B## and ##LC(A\cup B)## related?
    Then given any submodule ##C \subseteq M## which contains both, ##A## and ##B##, does it contain ##LC(A\cup B)##? And why?

    By theorem 2.1 ##A\cap B## is a submodule of ##M##. Then how are ##A\cap B## and ##LC(A\cap B)## related?
    Now given any submodule ##C##, that is contained in ##A## as well as in ##B##, is it contained in ##A\cap B## or ##LC(A\cap B)##? And why?

    And at last: If you know all this about any possible submodule ##C \subseteq M##, what does it tell you?
     
  4. Aug 16, 2016 #3

    andrewkirk

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    Proof of Q1.
    Define [itex]A+B\equiv \{a+b\ :\ a\in A\wedge b\in B\}[/itex]. This is easily shown to be closed under addition and scalar multiplication, and hence a module. It must be a subset of [itex]M[/itex] because [itex]M[/itex] is closed under addition and scalar multiplication and every element of [itex]A+B[/itex] can be expressed as the addition or scalar, multiplication of elements of [itex]A[/itex] and/or [itex]B[/itex], which are elements of [itex]M[/itex].

    Let ##G_A## and ##G_B## be generating sets for ##A## and ##B## respectively. Then, by the definition of [itex]A+B[/itex], [itex]G_A\cup G_B[/itex] is a generating set for [itex]A+B[/itex].

    Let [itex]C[/itex] be a submodule of [itex]M[/itex] that contains [itex]A[/itex] and [itex]B[/itex]. Then [itex]C[/itex] contains both [itex]G_A[/itex] and [itex]G_B[/itex] and hence contains [itex]G_A\cup G_B[/itex]. Any element of [itex]A+B[/itex] is a finite linear combination of elements of [itex]G_A\cup G_B[/itex] and, since that union is in [itex]C[/itex] which, being a module, is closed under finite linear combinations, the element must also be in [itex]C[/itex]. So we have [itex]A+B\subseteq C[/itex].

    I imagine the proof for Q2 may be similar but using intersections of generating sets rather than unions. Why not have a go at that and let me know if you get stuck.

    Andrew
     
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