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