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 Post subject: Conditions That Imply CommutativityPosted: Thu Jun 09, 2016 9:29 am
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Joined: Tue Nov 10, 2015 8:25 pm
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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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 Post subject: Re: Conditions That Imply CommutativityPosted: Sun Jun 26, 2016 7:58 am
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Joined: Mon Nov 09, 2015 1:36 am
Posts: 460
Location: Ioannina, Greece
2. is answered here: Commutative ring

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Grigorios Kostakos

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