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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. \( \displaystyle \forall r \in R : r^3 - r \in Z(R) \)

2. is answered here: Commutative ring