Answer to Question #16539 in Algebra for Tsit Lam

Question

Let p be a fixed prime. Show that any ring (with identity) of order p^2 is commutative

Answer 1

If the additive order of 1 is p2, then R = Z · 1 is clearly commutative. If the additive order of 1 is p, then R = Z · 1 ⊕Z · a for any a∈Z · 1, and again R is commutative.