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Multiplicative and additive identities as successors

  1. Jul 2, 2012 #1
    Fact: The ring of integers Z is totally ordered: for any distinct elements a and b in Z, either a>b or a<b.

    Fact: The ring of integers is discrete, in the sense that for any element a in Z, there exists an element b in Z such that there is no element c in Z with a<c<b, and the same argument holds with the greater than signs flipped. In other words, successors and predecessors exist in Z (but not in R, for example).

    Observation: In Z, the multiplicative identity 1 is the successor of the additive identity 0.

    Questions: Is this fact a coincidence? Does this have any significance? Must the multiplicative identity always succeed the additive identity in rings that have the same properties as Z, assuming there are any other?
     
  2. jcsd
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