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Math Help - Ring with identity

  1. #1
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    Ring with identity

    Let x be an element in a ring with identity, and suppose x^{n} = 0 for some positive integer n.
    Now i want to proof that (1+x) is a unit.

    I tried to use (1+x)(1-x+x^2-x^3+...) =1 but i can't solve it. Any hints?
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  2. #2
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    suppose n is odd.
    then (-1)^n+1=0 so x+1 divides 1+x^n=1+0=1 by factor theorem
    so there is an element y such that (1+x)(y)=1

    what if n i even?

    so after you have done this you can use long division for each aswer to give you the unique y needed and you see that the elements are both of the form you wrote them in and is composed of the identity element and x, and so y is in the ring with identity.
    Last edited by Krahl; March 2nd 2010 at 03:34 AM.
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  3. #3
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    Quote Originally Posted by bram kierkels View Post
    Let x be an element in a ring with identity, and suppose x^{n} = 0 for some positive integer n.
    Now i want to proof that (1+x) is a unit.

    I tried to use (1+x)(1-x+x^2-x^3+...) =1 but i can't solve it. Any hints?

    You have the idea. Use the well-known identity from geometric sequences:

    \frac{1-x^n}{1-x}=1+x+x^2+\ldots+x^{n-1}\Longrightarrow 1-x^n=(1-x)(1+x+\ldots+x^{n-1}) , and change conveniently the sign if you will (though you don't have

    to, since x^n=0\Longleftrightarrow (-x)^n=0 )

    Tonio
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