Maximal ideals are prime

Maximal ideals are prime: Let $R$ be a commutative ring with $1$, and let $\mathfrak m$ be a maximal ideal of $R$. Then $\mathfrak m$ is a prime ideal .

A standard proof uses the characterization that $R/\mathfrak m$ is a field and hence an integral domain, and translates the domain property back to primeness via the quotient ring .