Maximal ideal iff quotient is a field

Maximal ideal iff quotient is field: Let $R$ be a commutative ring with $1$, and let $I\triangleleft R$ be an ideal . Then $I$ is a maximal ideal if and only if the quotient ring $R/I$ is a field .

This equivalence is the main bridge between ideal-theoretic maximality and “no nontrivial quotients” in commutative algebra.