Is "Multiplication Commutative in Rings?

[1MileCrash]

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.

 

[1MileCrash]

Never mind, I have it.

But how can I show that -1 * -1 = 1 where 1 is the multiplicative identity?

 

[lurflurf]

Use the distributive property with
(-1)(1+(-1))=0

 

[1MileCrash]

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]

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]

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.