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Let $R$ be a finite ring of order $p^2$ with unity $e$ and characteristic $p$. This ring is commutative but I cannot get why it is.

I know that this ring looks as $\mathbb Z /p\mathbb Z \times Z /p\mathbb Z$. Could anyone help me to show the commutativity of this ring?

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migrated from Jul 25 '13 at 20:47

This question came from our site for professional mathematicians.

marked as duplicate by Jack Schmidt, Pete L. Clark, Dan Rust, Danny Cheuk, user1551 Jul 25 '13 at 21:14

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

This has been answered already here:…, and here:… – Dietrich Burde Jul 25 '13 at 20:52

Pick an arbitrary element $x \in R$ which does not belong to the subfield $\mathbb{F}_p=\{k.e \mid k \in \mathbb{Z}\}$, and note that $R=\mathbb{F}_p \oplus \mathbb{F}_p x$.

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