Prime ideal iff quotient is an integral domain

Prime ideal iff quotient is integral domain: Let $R$ be a commutative ring with $1$, and let $P\triangleleft R$ be an ideal . Then $P$ is a prime ideal if and only if the quotient ring $R/P$ is an integral domain .

This is the fundamental translation between multiplicative primeness and “domain-like” behavior after quotienting.