# Quick ring question

by 1MileCrash
Tags: ring
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 P: 1,291 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.
 P: 1,291 Never mind, I have it. But how can I show that -1 * -1 = 1 where 1 is the multiplicative identity?
 HW Helper P: 2,263 Use the distributive property with (-1)(1+(-1))=0
P: 1,291
Quick ring question

 Quote by lurflurf Use the distributive property with (-1)(1+(-1))=0
Cool,

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

(-1)(-1) = 1 by definition
Mentor
P: 21,214
 Quote by 1MileCrash Cool, (-1)(1) + (-1)(-1) = 0 -1 + (-1)(-1) = 0 (-1)(-1) = 1 by definition
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.
P: 1,291
 Quote by Mark44 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.
Yes, exactly.

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