There is no commutative ring whose group of units is isomorphic to $\mathbb{Z}/p^2\mathbb{Z}$

Question

I tried to show that there is no commutative ring with the unit whose group of units is isomorphic to $\mathbb{Z}/p^2\mathbb{Z}$. I can show if the ring is a finite field,it is easy. But I can no idea about general case. Please give me some advice.