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Thread: Ring [extended level] part 2 :D

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

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

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

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

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