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Thread: module concept

  1. #1
    Nov 2012

    module concept

    I thought that I understand the module concept but I've been confused since I had to deal with such task:
    "Find an example of finitely generated module that is non-finitely generated abelian group."

    Is that possible? thanks in advance for your help
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  2. #2
    MHF Contributor

    Mar 2011

    Re: module concept

    consider the module over the rational numbers (R = Q) generated by 1: that is the module Q itself (since Q being a ring, is a module over Q).

    if Q was a finitely-generated module over Z (abelian groups are Z-modules), we would be able to express any rational number as a Z-linear combination of a finite set of rationals:

    B = {q1,...,qn}.

    write qj = aj/bj, where aj,bj ≠ 0, and gcd(aj,bj) = 1.

    consider k = lcm(b1,...,bn).

    certainly any Z-linear combination of elements of B can be written as N/k, where N is some integer (which N we get depends on which Z-linear combination we have, but we can use the same k).

    taking out common factors, we have N/k = N'/k', where gcd(N',k') = 1.

    let p be a prime number that does not divide k (we can ALWAYS find such a prime number, since there are infinitely many primes, but k has only a finite number of prime factors).

    i claim 1/p is not in span(B) (where this indicates the Z-linear span).

    for if it were, we have 1/p = N'/k', and thus: k' = N'p, hence p divides k', and thus divides k, contradiction.
    Thanks from snakus
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