Prove if a ring has a unity, then it is unique

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In summary, the conversation discusses a proof that if a ring has a unity, then it is unique. The proof involves assuming the existence of two distinct unities in a ring and using the definition of a unity to show that they must be equal. The problem arises in the case of an integral domain, but it is resolved by showing that both unities must satisfy the same equation, leading to the conclusion that they are equal. The conversation concludes with a clarification and gratitude for the help.
  • #1
SomeRandomGuy
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Prove if a ring has a unity, then it is unique:

Here is what I have so far:
Proof: Assume there exists a ring R that contains two distinct unity's, call a and b, where a != b. By the definition of a unity, we get ax = xa = x and bx = xb = x for all x != 0 in R. So, ax = xa = bx = xb = x. If the ring is an integral domain, we get a = b because there are no zero divisors.

Problem occurs for the case of an integral domain. Thanks for help.
 
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  • #2
This might be not work, since I don't remember all the definitions off-hand... But if a and b are both unities in some ring R, then ab = a and ab = b. Hence a = b.
 
  • #3
Muzza said:
This might be not work, since I don't remember all the definitions off-hand... But if a and b are both unities in some ring R, then ab = a and ab = b. Hence a = b.

I believe that's what we are trying to prove. If a and b are both unity's in a ring, then a = b.
 
  • #4
Apparently my deduction wasn't explicit enough, or you looked past one of the sentences in my post.

Since a was a unity in R, ax = x for all x in R. In particular, it must hold if we set x = b. Hence ab = b.

But b was also a unity in R, so that xb = x for all x in R. In particular, it must hold if we set x = a, so that ab = a.

So we have proved that ab = a and ab = b. By transitivity, we must have that a = b.
 
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  • #5
Muzza said:
Apparently my deduction wasn't explicit enough, or you looked past one of the sentences in my post.

Since a was a unity in R, ax = x for all x in R. In particular, it must hold if we set x = b. Hence ab = b.

But b was also a unity in R, so that xb = x for all x in R. In particular, it must hold if we set x = a, so that ab = a.

So we have proved that ab = a and ab = b. By transitivity, we must have that a = b.

Your right. I didn't even read the sentence ab = a and ab = b. My bad, sorry for the confusion and tanks very much for the help.
 

1. What is a unity in a ring?

A unity in a ring is an element that acts as the identity element for multiplication. In other words, when this element is multiplied with any other element in the ring, the result is the other element itself.

2. How can you prove that a ring has a unique unity?

To prove that a ring has a unique unity, we need to show that there is only one element in the ring that satisfies the definition of a unity. This can be done by assuming the existence of two unity elements and then using the properties of a ring to show that they must be equal.

3. Can a ring have more than one unity?

No, a ring can only have one unity. This is because if there were two unity elements, say a and b, then a*b would also be a unity element. However, by the definition of a unity, a*b must equal both a and b, which means a and b are equal. Therefore, there can only be one unity element in a ring.

4. What happens if a ring does not have a unity?

If a ring does not have a unity, then it is called a non-unital ring. These types of rings do not have an identity element for multiplication, which means that the properties of a unity do not hold. Non-unital rings are still valid mathematical structures, but they behave differently than rings with a unity.

5. How does the uniqueness of a unity in a ring affect its properties?

The uniqueness of a unity in a ring is an important property because it allows us to use the unity as a reference point for other operations in the ring. For example, we can use the unity to define the inverse of an element in the ring. Additionally, the uniqueness of a unity also helps us prove other properties of rings, such as the existence of a multiplicative inverse for every non-zero element.

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