Prove that prime ideals of a finite ring are maximal

Question

Let $R$ be a finite commutative unitary ring. How to prove that each prime ideal of $R$ is maximal?

Answer 1

Let $\mathfrak{p}$ be a prime ideal in $R$. Then $R/\mathfrak{p}$ is a finite integral domain, thus it is a field, hence $\mathfrak{p}$ is maximal.

Answer 2

Let $R$ be commutative. We know that any maximal ideals is prime. Conversely, for any prime ideal $P$ of $R$, the quotient ring $R/P$ is a finite integral domain, so it is a field. Then in commutative finite rings, prime ideals are the same with maximal ideals.