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Quick ring question

by 1MileCrash
Tags: ring
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1MileCrash
#1
Mar31-14, 04:42 PM
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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.
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1MileCrash
#2
Mar31-14, 04:50 PM
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Never mind, I have it.

But how can I show that -1 * -1 = 1 where 1 is the multiplicative identity?
lurflurf
#3
Mar31-14, 07:27 PM
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Use the distributive property with
(-1)(1+(-1))=0

1MileCrash
#4
Mar31-14, 08:29 PM
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Quick ring question

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

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

(-1)(-1) = 1 by definition
Mark44
#5
Mar31-14, 08:49 PM
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Quote Quote by 1MileCrash View Post
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.
1MileCrash
#6
Mar31-14, 10:11 PM
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Quote Quote by Mark44 View Post
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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