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Math Help - Module Terminology

  1. #1
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    Module Terminology

    I'm having trouble wrapping my head around notions of free modules and here's why (quoting from the book):
    So let M be a unital R-module (R commutative with identity). Suppose also that M is finitely generated, say M = <m_{1},...,m_{n}>. M is said to be freely generated by m_{1},...,m_{n} if
    r_{1}m_{1}+...+r_{n}m_{n} = 0 \Rightarrow r_{1}=...=r_{n}=0
    Furthermore, M is called free if it is freely generated by a finite set.

    I'm confused as to what assumptions are being made all over the place, so here are my questions (I'll always assume I'm talking about unital modules over commutative rings with 1)
    1) The way the notion of freely generated is presented makes it seem that if I say "M is an R-module freely generated by m_{1},...,m_{n}" then I'm implicitly assuming it is finitely generated. Is this true? Can it happen that M is not finitely generated but freely generated? If not then:
    2) If M is freely generated by m_{1},...,m_{n}, can it happen that M is generated by more or less elements? If not then:
    3) This condition looks suspiciously familiar to linear independence in vector spaces, but terms like "linear dependence" and "basis" are never used; the book even says that free modules behave very much like vector spaces. Is it wrong to think of this as the analogous of linear independent sets in vector spaces?

    Any clarification is very welcome!

    EDIT: the reason I'm asking is because right after there is a theorem stating M is freely generated by m_{1},...,m_{n} if and only if M can be uniquely expressed by finite sums of m_{1},...,m_{n}. The proof is short and vague so I was trying to fill in the details but that brings me to question 1) above. Is this theorem just stating 1 result (namely that the representation of elements is unique) or 2 (that there IS a representation of the elements AND furthermore it's unique)? Or rather: is the fact that M = <m_{1},...,m_{n}> by definition of freely generated or a consequence of it?
    Last edited by bleys; October 22nd 2010 at 06:26 AM.
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  2. #2
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    Quote Originally Posted by bleys View Post
    I'm having trouble wrapping my head around notions of free modules and here's why (quoting from the book):
    So let M be a unital R-module (R commutative with identity). Suppose also that M is finitely generated, say M = <m_{1},...,m_{n}>. M is said to be freely generated by m_{1},...,m_{n} if
    r_{1}m_{1}+...+r_{n}m_{n} = 0 \Rightarrow r_{1}=...=r_{n}=0
    Furthermore, M is called free if it is freely generated by a finite set.

    I'm confused as to what assumptions are being made all over the place, so here are my questions (I'll always assume I'm talking about unital modules over commutative rings with 1)
    1) The way the notion of freely generated is presented makes it seem that if I say "M is an R-module freely generated by m_{1},...,m_{n}" then I'm implicitly assuming it is finitely generated. Is this true? Can it happen that M is not finitely generated but freely generated?

    Certainly: the \mathbb{Z}-module \oplus\limits^\infty_{i=1}Z_i\,,\,\,Z_i:=\mathbb{Z  }\,,\,\forall i is infinitely generated and free.


    If not then:
    2) If M is freely generated by m_{1},...,m_{n}, can it happen that M is generated by more or less elements? If not then:
    3) This condition looks suspiciously familiar to linear independence in vector spaces, but terms like "linear dependence" and "basis" are never used; the book even says that free modules behave very much like vector spaces. Is it wrong to think of this as the analogous of linear independent sets in vector spaces?

    Any clarification is very welcome!

    EDIT: the reason I'm asking is because right after there is a theorem stating M is freely generated by m_{1},...,m_{n} if and only if M can be uniquely expressed by finite sums of m_{1},...,m_{n}. The proof is short and vague so I was trying to fill in the details but that brings me to question 1) above. Is this theorem just stating 1 result (namely that the representation of elements is unique) or 2 (that there IS a representation of the elements AND furthermore it's unique)? Or rather: is the fact that M = <m_{1},...,m_{n}> by definition of freely generated or a consequence of it?
    Take a peek at some good algebra book, like "Abstract Algebra" by Dummit&Foote, por "algebra" by Hungerford, etc.

    Tonio
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  3. #3
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    Why is the module you gave me free? If I go by the definition given, it means it is freely generated by a finite set. Does that mean I have to look at finite subsets of the generating set and show they are linearly independent?
    From other sources I understand 'free' means the module has a linearly independent generating set. I'm having trouble understanding what 'freely generated' means; is it just a generalization of 'free' to include the infinite case? Are they equivalent? I'm thinking some (most I'm guessing) authors use 'free' to include also infinitely generated...
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  4. #4
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    Quote Originally Posted by bleys View Post
    Why is the module you gave me free? If I go by the definition given, it means it is freely generated by a finite set. Does that mean I have to look at finite subsets of the generating set and show they are linearly independent?
    From other sources I understand 'free' means the module has a linearly independent generating set. I'm having trouble understanding what 'freely generated' means; is it just a generalization of 'free' to include the infinite case? Are they equivalent? I'm thinking some (most I'm guessing) authors use 'free' to include also infinitely generated...
    It seems like the definition you have only talks about fin. generated free module, but just like with vector space we can talk of infinitely free generated.

    In the example I gave you the set \{(1,0,0,...), (0,1,0,...),...\} is a free \mathbb{Z}-generating set of that module (do the maths, don't just ask why),

    but first do read a little about free modules in some good algebra book.

    Tonio
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  5. #5
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    Well I did understand what the generating set of the module is and that it's lin. ind., but I can't very well do the maths when it's not clear what the definition of free is, can I?! Never mind that it hasn't been clarified what 'freely generated' means...
    From what I've read then free iff freely generated
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  6. #6
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    Quote Originally Posted by bleys View Post
    Well I did understand what the generating set of the module is and that it's lin. ind., but I can't very well do the maths when it's not clear what the definition of free is, can I?!


    Indeed, and here is where the "read this stuff in a good algebra book" part fits in, though in your first message you already mentioned it.

    The only thing left is to read the definition for "free infinitely generated ", which is very similar to infinite dimension in linear algebra:

    Every finite subset of the generating set must be free in the sense you wrote in your first message.

    Tonio



    Never mind that it hasn't been clarified what 'freely generated' means...
    From what I've read then free iff freely generated
    .
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  7. #7
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    Alright, thank you for your help Tonio.
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