Is "Multiplication Commutative in Rings?

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In summary, the conversation discusses whether the equation (-x) * y = x * (-y) is true for all rings, and how to prove that -1 * -1 = 1 using the distributive property and the definition of additive inverses. It is concluded that -1(-1) must equal 1.
  • #1
1MileCrash
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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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  • #2
Never mind, I have it.

But how can I show that -1 * -1 = 1 where 1 is the multiplicative identity?
 
Last edited:
  • #3
Use the distributive property with
(-1)(1+(-1))=0
 
  • #4
lurflurf said:
Use the distributive property with
(-1)(1+(-1))=0
Cool,

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

(-1)(-1) = 1 by definition
 
  • #5
1MileCrash said:
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.
 
  • #6
Mark44 said:
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.
 

1. Is multiplication commutative in all rings?

No, multiplication is not always commutative in rings. In general, commutativity is not a requirement for rings, so there are many examples of rings where multiplication is not commutative.

2. What is an example of a ring where multiplication is not commutative?

An example of a ring where multiplication is not commutative is the ring of 2x2 matrices with real entries. In this ring, the order of multiplication matters, so it is not commutative.

3. How does commutativity of multiplication affect the properties of a ring?

If multiplication is commutative in a ring, it means that the order of multiplication does not matter. This can simplify calculations and make certain properties, such as the distributive property, easier to work with.

4. Can commutativity of multiplication change in different rings?

Yes, the commutativity of multiplication can change in different rings. Some rings have commutative multiplication, while others do not. It depends on the specific structure and properties of the ring.

5. What are some applications of commutativity in ring theory?

Commutativity of multiplication in rings has applications in various areas of mathematics, including abstract algebra, number theory, and algebraic geometry. It also has practical applications in fields such as coding theory and cryptography.

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