Universal property of quotient rings

Universal property of quotient rings: Let $R$ be a ring, let $I\triangleleft R$ be an ideal , and let $\pi:R\to R/I$ be the canonical projection onto the quotient ring . For any ring homomorphism $f:R\to S$ such that $I\subseteq \ker(f)$ (where $\ker(f)$ is the kernel ), there exists a unique ring homomorphism $\bar f:R/I\to S$ with

This property characterizes $R/I$ up to unique isomorphism and is the categorical mechanism behind “imposing relations” by quotienting.