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Math Help - Irreduciple element -problem

  1. #1
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    Irreduciple element -problem

    I have this kind of exercise in which I need help:

    Let R be integral domain. Assume that there is an element p \in R\setminus U(R), p \neq 0, so that if p\mid xy then  p\mid x or  p\mid y with every x,y \in R. Show that p is irreducible. (U(R) = the group of units of R)

    Any help would be great.
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  2. #2
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    Quote Originally Posted by Ester View Post
    I have this kind of exercise in which I need help:

    Let R be integral domain. Assume that there is an element p \in R\setminus U(R), p \neq 0, so that if p\mid xy then  p\mid x or  p\mid y with every x,y \in R. Show that p is irreducible. (U(R) = the group of units of R)

    Any help would be great.


    Suppose p=ab\Longrightarrow p\mid ab\Longrightarrow p\mid a\,\,or\,\,p\mid b .

    WLTG suppose p\mid a\Longrightarrow a=px\Longrightarrow p = ab=pxb\Longrightarrow xb=1.

    End now the proof.

    Tonio
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