- \( \displaystyle \forall r \in R : r^2 = r \)
- \( \displaystyle \forall r \in R : r^3 = r \)
- \( \displaystyle \forall r \in R : r^2 - r \in Z(R) \)
- \( \displaystyle \forall r \in R : r^2 + r \in Z(R) \)
- \( \displaystyle \forall r \in R : r^3 - r \in Z(R) \)
Conditions That Imply Commutativity
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Conditions That Imply Commutativity
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:
- Grigorios Kostakos
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- Location: Ioannina, Greece
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