Quick ring question

  1. Is (-x) * y = x * (-y) true for all rings?

    It seems simple enough but I feel like * must be commutative when trying to prove this.
     
  2. jcsd
  3. Never mind, I have it.

    But how can I show that -1 * -1 = 1 where 1 is the multiplicative identity?
     
    Last edited: Mar 31, 2014
  4. lurflurf

    lurflurf 2,325
    Homework Helper

    Use the distributive property with
    (-1)(1+(-1))=0
     
  5. Cool,

    (-1)(1) + (-1)(-1) = 0
    -1 + (-1)(-1) = 0

    (-1)(-1) = 1 by definition
     
  6. Mark44

    Staff: Mentor

    Not by definition.

    1 + (-1) = 0 since 1 and -1 are additive inverses of each other
    -1(1 + (-1)) = -1(0) = 0, since 0 times anything is 0.
    -1(1) + (-1)(-1) = 0
    Since -1(1) and (-1)(-1) add to zero, they are additive inverses.
    We know that -1(1) = -1, since 1 is the multiplicative identity,
    so -1(-1) must equal 1.
     
  7. Yes, exactly.
     
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