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Math Help - Two linear algebra questions (finitely generated, isomorphism)

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    Two linear algebra questions (finitely generated, isomorphism)

    I have two exercises which are driving me crazy.

    1) Is Q finitely generated Z-module?

    2) Let d,e \in N/\{0\}. Show, that Hom_Z(Z_d,Z_e) \cong Z_f, where f = gcd (d,e).
    I have previously proved that Hom_Z(Z_d, G) \cong \{x \in G | dx = 0\}, where G is an Abelian group and  d \in N/\{0\}, so it's enough to show that H:=\{\overline{n} \in Z_e | d\overline{n} = 0\} \cong Z_f. And I have been given the rule \overline{k} \mapsto \overline{k(e/f)}. So I have to show that f: Z_f \rightarrow H, f(\overline{k}) = \overline{k(e/f)}, is isomorphism.

    I have been able to show that this function is homomorphism and injective, but the problem here is how I can show that this function is surjective?
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    Quote Originally Posted by Ester View Post
    I have two exercises which are driving me crazy.

    1) Is Q finitely generated Z-module?

    2) Let d,e \in N/\{0\}. Show, that Hom_Z(Z_d,Z_e) \cong Z_f, where f = gcd (d,e).
    I have previously proved that Hom_Z(Z_d, G) \cong \{x \in G | dx = 0\}, where G is an Abelian group and  d \in N/\{0\}, so it's enough to show that H:=\{\overline{n} \in Z_e | d\overline{n} = 0\} \cong Z_f. And I have been given the rule \overline{k} \mapsto \overline{k(e/f)}. So I have to show that f: Z_f \rightarrow H, f(\overline{k}) = \overline{k(e/f)}, is isomorphism.

    I have been able to show that this function is homomorphism and injective, but the problem here is how I can show that this function is surjective?
    For the first one, take a finite set of rational numbers and look at the \mathbb{Z}-submodule which they generate. What are the possible denominators of its elements?
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  3. #3
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    Quote Originally Posted by Bruno J. View Post
    For the first one, take a finite set of rational numbers and look at the \mathbb{Z}-submodule which they generate. What are the possible denominators of its elements?
    Okey... I have no idea, but I came up something like this:

    S = \{x_1, ... , x_n\} where x_1, ... , x_n \in Q. So S \subset Q. Z -submodule, which they generate, is <S>, right? And if a_1, ... , a_n \in Z and x_1, ... , x_n \in S, then because S \subset <S> and <S> is submodule, a_1x_1 + ... + a_nx_n \in <S>.

    Am I even close?
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    Quote Originally Posted by Ester View Post
    Okey... I have no idea, but I came up something like this:

    S = \{x_1, ... , x_n\} where x_1, ... , x_n \in Q. So S \subset Q. Z -submodule, which they generate, is <S>, right? And if a_1, ... , a_n \in Z and x_1, ... , x_n \in S, then because S \subset <S> and <S> is submodule, a_1x_1 + ... + a_nx_n \in <S>.

    Am I even close?

    You´re close, in fact on it, to the definition of Z-module but not to show that the finitely generated Z-submodule S of Q cannot be the hole Q...
    Read carefully what Bruno wrote you: in any Z-combination a_1z_1+\ldots+a_nz_n of the elements of S, how many

    posible denominators can there be? How many primes are there?...
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    Quote Originally Posted by tonio View Post
    You´re close, in fact on it, to the definition of Z-module but not to show that the finitely generated Z-submodule S of Q cannot be the hole Q...
    Read carefully what Bruno wrote you: in any Z-combination a_1z_1+\ldots+a_nz_n of the elements of S, how many

    posible denominators can there be? How many primes are there?...
    Well... How many primes? Infinite amount.
    Does this mean that if I choose an arbitrary k \in N and an arbitrary a_1x_1+ ... + a_nx_n \in <S>, then (a_1x_1+ ... + a_nx_n)/k \in Q but it doesn't belong in <S>?
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    Quote Originally Posted by Ester View Post
    Well... How many primes? Infinite amount.
    Does this mean that if I choose an arbitrary k \in N and an arbitrary a_1x_1+ ... + a_nx_n \in <S>, then (a_1x_1+ ... + a_nx_n)/k \in Q but it doesn't belong in <S>?

    I don't understand what you meant1) how many possible primes can divide ANY denominator in a linear combination

    a_1x_2+\ldots+a_nx_n\in S , with a_i\in\mathbb{Z} ? After you answer this question pass on to the infinitude of primes...

    Tonio
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    I came up something like this:

    Let's choose arbitrary s \in <S>, so s = a_1x_1 + ... + a_nx_n, where a_i \in Z and let's choose w \in N/\{0,1\} so that gcd(w,x_i) = 1. Now s/w \in Q.

    Antithesis: s/w \in <S>.
    s/w = (a_1x_1 + ... + a_nx_n)/w = (a_1x_1)/w + ... + (a_nx_n)/w
    Now a_1/w, ... , a_n/w \in Z iff w=1. But this is contradiction. So s/w does not belong in <S>.
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    Quote Originally Posted by Ester View Post
    I came up something like this:

    Let's choose arbitrary s \in <S>, so s = a_1x_1 + ... + a_nx_n, where a_i \in Z and let's choose w \in N/\{0,1\} so that gcd(w,x_i) = 1. Now s/w \in Q.

    Antithesis: s/w \in <S>.
    s/w = (a_1x_1 + ... + a_nx_n)/w = (a_1x_1)/w + ... + (a_nx_n)/w
    Now a_1/w, ... , a_n/w \in Z iff w=1. But this is contradiction.

    No contradiction at all: w was chosen to be pairwise coprime with the x_i's , not with the a_i\s ...

    You insist in not paying attention to the hint you've been given already three times...

    Tonio



    So s/w does not belong in <S>.
    .
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  9. #9
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    Okey.

    I'm not paying attention to the hint, because I don't get it.
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  10. #10
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    Quote Originally Posted by Ester View Post
    Okey.

    I'm not paying attention to the hint, because I don't get it.

    1) How many different primes can be part of any of the denominators of \{x_1,\ldots,x_n\} ?

    2) If  a_i\in\mathbb{Z} , can the number of primes in the denominator of a_1x_1+\ldots+a_nx_n be different from the number in (1)?

    3) Since there are INFINITE primes then...

    Tonio
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