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When do objects factorize uniquely?

  1. Jan 5, 2009 #1
    Integer number and groups have unique factorizations into irreducible parts.

    In general, what are the abstract requirements for mathematical objects to factorize uniquely?

    I suppose for this question to ask a binary operation has to be defined. So is this question only possible for elements of groups?
     
  2. jcsd
  3. Jan 5, 2009 #2

    CRGreathouse

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    The specific term for this (more specific than group) is UFD: a unique factorization domain. I don't know of any easy characterizations of UFDs.
     
  4. Jan 7, 2009 #3
    Isn't it basically just a case of unique factorization in an integral domain/communitive ring, similar to what is found with the integers, and called, "The Fundamental Theorem of Arithmetic"? A prime being a number, not 1, divisible only by 1 or by itself.

    The main property of a prime being if p/ab, then p/a or p/b. This can be shown with the Euclidean Algorithm.
    Proof: if p does not divide a, then there exists integers such that: 1 = pm+an. Thus b=bpm+ban. Thus p divides b.

    Gerenuk: I suppose for this question to ask a binary operation has to be defined. So is this question only possible for elements of groups?

    Every element of a group has an inverse. In such a case a divides b gives b times a inverse. What we want is a group under addition and multiplicatively an Integral Domain.
     
    Last edited: Jan 7, 2009
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