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- Jun 20, 2014

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Let $R$ be a ring such that for all $r\in R$, $r^3 = r$. Prove $R$ is commutative.

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- Thread starter
- Moderator
- #1

- Jun 20, 2014

- 1,896

-----

Let $R$ be a ring such that for all $r\in R$, $r^3 = r$. Prove $R$ is commutative.

-----

Remember to read the POTW submission guidelines to find out how to submit your answers!

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- Jun 20, 2014

- 1,896

Now given $a,b\in R$, we have

$$ab = a^3b = a^2(ab) = aba^2 = ab^3a^2 = (ab)b^2a^2 = b^2aba^2 = b(ba)^2a = (ba)^2ba = (ba)^3 = ba$$

Hence, $R$ is commutative.

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