Quotient Ring and finite fields

Question

How is a quotient ring $\mathbb Z/p^e\mathbb Z$ (where p is prime and $e>2$) different from a finite field $\mathbb F_{p^e}$? When they are both rings, have the same elements?

I thought a finite field p is generated by $\mathbb Z/p\mathbb Z$?

Answer 1

The ring $\mathbb Z/p^e\mathbb Z$ with $e\ge2$ is not a field because it has divisors of zero. Specifically, the product of $p+p^e\mathbb Z$ and $p^{e-1}+p^e\mathbb Z$ is zero. The additive group of $\mathbb F_{p^e}$ is isomorphic to $(\mathbb Z/p\mathbb Z)^e$, whereas that of $\mathbb Z/p^e\mathbb Z$ is cyclic.