Conditions That Imply Commutativity
Posted: Thu Jun 09, 2016 9:29 am
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:
- \( \displaystyle \forall r \in R : r^2 = r \)
- \( \displaystyle \forall r \in R : r^3 = r \)
- \( \displaystyle \forall r \in R : r^2 - r \in Z(R) \)
- \( \displaystyle \forall r \in R : r^2 + r \in Z(R) \)
- \( \displaystyle \forall r \in R : r^3 - r \in Z(R) \)