# Commutativity of ring of order $p^2$ with unity $e$ and characteristic $p$ [duplicate]

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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?

## marked as duplicate by Jack Schmidt, Pete L. Clark, Dan Rust, user67258, user1551Jul 25 '13 at 21:14

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## migrated from mathoverflow.netJul 25 '13 at 20:47

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## 1 Answer

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 $R=\mathbb{F}_p \oplus \mathbb{F}_p x$.