Number of elements in a ring with identity.

In summary, In summary, The ring R has two elements, -1 and 1, which are both their own inverses. This is because for any element x in R, x^2 = 1_R, and since 1_R is the identity element, this means that x is its own inverse. Therefore, there are only two elements in R.
  • #1
Stephen88
61
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Homework Statement


1_R=identity in the ring R.
/=...not equal
Having some issues with this any help will be great:
Let R be a ring with identity, such that
x^2 = 1_R for all 0_R /= x ,where x belongs to R. How many elements are in R?


Homework Equations





The Attempt at a Solution


I'm looking at the element x^2 +1_R which can be = 0 or /=0.
If it is the first case then x=-1 and where are done |R|=2 because of the powers of x.
If the latter then (x^2 +1_R)^2=1_R...after calculation this gives x=-1/2 which is false since x^2=1 .
 
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  • #2
never mind
 
Last edited:

1. What does the "identity" in a ring refer to?

The identity in a ring refers to an element that, when combined with any other element in the ring using the ring's operation, results in the other element unchanged. For example, in a ring of integers under addition, the identity is 0, since any integer added to 0 remains unchanged.

2. Can a ring have more than one identity?

No, a ring can only have one identity element. This is because if there were two distinct identity elements, say a and b, then a*b would be equal to both a and b, making them the same element.

3. Is the identity element always required in a ring?

Yes, the identity element is a necessary component of a ring. Without it, the ring would not have a starting point for its operation and would not satisfy the axioms that define a ring.

4. How does the number of elements in a ring with identity compare to the number of elements in a ring without identity?

The number of elements in a ring with identity is always one more than the number of elements in a ring without identity. This is because the identity element is unique and must be included in the count of elements for a ring with identity.

5. Can the identity element be any element in a ring?

No, the identity element must have specific properties in order to be considered as such. In a ring, the identity element must be both a left and right identity, meaning that it must satisfy the equations a*e = a and e*a = a for all elements a in the ring, where e is the identity element.

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