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SSC Pure Mathematics QUESTION #1258
Question 1
Let \(R\) be a ring where \(x^2 = x\) for all \(x\in R\) (Boolean ring). Then \(R\) must be:
  • A field
  • Commutative✔️
  • An integral domain
  • Has characteristic 3
Correct Answer Explanation
If \(x^2 = x\) for all \(x\in R\), then for any \(a,b\in R\): \((a+b)^2 = a+b\), expanding gives \(a^2+ab+ba+b^2 = a+b\), so \(a+ab+ba+b = a+b\), hence \(ab+ba=0\), i.e. \(ab = -ba\). Also, \(2x=0\) for all \(x\) (characteristic 2), so \(-1=1\) and \(ab=ba\). Thus \(R\) is commutative.