Non-isomorphicity of $(\mathbb{Z}/p^n\mathbb{Z})(x)$ and $(\mathbb{Z}/p\mathbb{Z})(x_1)(x_2)...(x_n)$

Question

How does one go about proving that $(\mathbb{Z}/p^n\mathbb{Z})(x) \not\cong (\mathbb{Z}/p\mathbb{Z})(x_1)(x_2)...(x_n)$ as rings?

Intuitively, I understand, but I am not sure how to make it concrete. I expect that I should assume that an isomorphism exists, but I am not sure from where I would derive a contradiction.