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PostPosted: Thu Jun 09, 2016 9:29 am 
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Let \( \displaystyle R \) be an associative ring with unity \( \displaystyle 1_{R} \). Show that each of the following conditions imply that \( \displaystyle R \) is commutative:

  1. \( \displaystyle \forall r \in R : r^2 = r \)
  2. \( \displaystyle \forall r \in R : r^3 = r \)
  3. \( \displaystyle \forall r \in R : r^2 - r \in Z(R) \)
  4. \( \displaystyle \forall r \in R : r^2 + r \in Z(R) \)
  5. \( \displaystyle \forall r \in R : r^3 - r \in Z(R) \)


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PostPosted: Sun Jun 26, 2016 7:58 am 
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2. is answered here: Commutative ring

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