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Math Help - Ring [extended level] part 2 :D

  1. #1
    Newbie
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    Talking Ring [extended level] part 2 :D

    (1)
    Given Integral domain R and a,b\in R\ \{0\}
    a) a\in R not unit, is called prime element
    if a|bc implies a|b or a|c.
    Show that a is prime element iff <a> is prime ideal
    b) a\in R not unit, is said un-reduced
    if a = bc implies b is unit or c is unit.
    Show that if p is prime and p|a_1...a_n \text{ then } p|a_i for some i.

    (2)
    If \mathbb{Z}[\sqrt{-3}]=\{a+b\sqrt{-3}|a,b\in\mathbb{Z}\}. What criteria must be held by 3\in\mathbb{Z}[\sqrt{-3}]? (is it prime or un-reduced?)
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  2. #2
    Junior Member
    Joined
    Mar 2010
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    (1a) Note that if p is prime then \langle p \rangle = \{ pr : r \in R\} = \{x \in R : p \mid x\}.

    (1b) You can show this by induction on n.

    (2) 3 = (\sqrt{-3})^2. Show that \sqrt{-3} is not a unit of \mathbb{Z}[\sqrt{-3}] and 3 \nmid \sqrt{-3} in \mathbb{Z}[\sqrt{-3}].
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