Let $R$ be a finite commutative unitary ring. How to prove that each prime ideal of $R$ is maximal?
Prove Prime Ideals of Finite Ring are Maximal – Abstract Algebra
abstract-algebrafinite-ringsmaximal-and-prime-idealsring-theory
abstract-algebrafinite-ringsmaximal-and-prime-idealsring-theory
Let $R$ be a finite commutative unitary ring. How to prove that each prime ideal of $R$ is maximal?
Best Answer
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.