Commutativity of ring of order $p^2$ with unity $e$ and characteristic $p$

Question

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?

This has been answered already here: http://math.stackexchange.com/questions/109506/classifying-unital-commutative-rings-of-order-p2, and here: http://math.stackexchange.com/questions/305512/ring-of-order-p2-is-commutative — Dietrich Burde, Jul 25 '13 at 20:52

score 1 · Answer 1 · answered Jul 25 '13 at 17:13

1

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

answered Jul 25 '13 at 17:13

Commutativity of ring of order $p^2$ with unity $e$ and characteristic $p$

1 Answers1