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Math Help - Ring with unity - (-1)^2 = 1

  1. #1
    Super Member Bernhard's Avatar
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    Ring with unity - (-1)^2 = 1

    Exercise 1 in section 7.1 of Dummit and Foote: Abstract Algebra is as follows:

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    Let R be a ring with 1

    Show that {(-1)}^2 = 1

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    The web site Project Crazy Project gives the following proof:

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    Let x \in R

    Then {(-1)}^2 x = (-1)(-x) = x

    Similarly x {(-1)}^2 = (-x)(-1) = x

    By the uniqueness of 1, {(-1)}^2 = 1

    ================================================== ===

    I am not sure why {(-1)}^2 x = (-1)(-x)

    I can see that by associativity of multiplication {(-1)}^2 x = (-1)[(-1)(x)]

    But then why is (-1)x = -x ... or is this just a formal way of expressing the same thing?

    Further why is (-1)(-x) = x?

    Peter
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  2. #2
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    Opalg's Avatar
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    Re: Ring with unity - (-1)^2 = 1

    Quote Originally Posted by Bernhard View Post
    But then why is (-1)x = -x?
    \begin{aligned}(-1)x + x &= (-1)x + (1)x\quad\text{(property of 1)} \\ &= (-1+1)x\quad\text{(distributive axiom)} \\ &= 0x\quad\text{(definition of 1)} \\ &= 0\quad\text{(property of 0)} \end{aligned}

    Thus (1)x is the negative of x, or in other words (1)x = x.

    Quote Originally Posted by Bernhard View Post
    Further why is (-1)(-x) = x?
    From the above result with x in place of x, it follows that (1)(x) = (x) = x.
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