Number of elements in a ring with identity.

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SUMMARY

The discussion centers on determining the number of elements in a ring R with identity, where every non-zero element x satisfies the equation x^2 = 1_R. The conclusion drawn is that if x^2 + 1_R = 0, then x must equal -1, leading to the result that the ring R contains exactly two elements, |R| = 2. The alternative case, where x^2 + 1_R ≠ 0, leads to contradictions, confirming the initial conclusion.

PREREQUISITES
  • Understanding of ring theory and the properties of rings with identity.
  • Familiarity with the concept of elements satisfying polynomial equations in algebra.
  • Knowledge of basic algebraic structures, specifically the definition of identity elements.
  • Ability to manipulate algebraic expressions and equations involving elements of a ring.
NEXT STEPS
  • Study the properties of rings with identity in abstract algebra.
  • Explore examples of finite rings and their structures.
  • Learn about polynomial equations in ring theory and their implications.
  • Investigate the classification of rings based on their elements and operations.
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Students of abstract algebra, mathematicians exploring ring theory, and educators teaching concepts related to algebraic structures.

Stephen88
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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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