AAR8

(Dis)prove: $\mathbb{Z}/(p)\times\mathbb{Z}/(p)\cong \mathbb{Z}/(p^2)\,$ as rings, where $p\,$ is prime.

This is false because $\mathbb{Z}/(p^2)\,$ has a zero divisor $p\cdot p=0$ but $\mathbb{Z}/(p)\times\mathbb{Z}/(p)\,$ doesn't because $p\,$ is prime.