Let $R$ and $S$ be unital rings with additive identities $0_R$ and $0_S$ and multiplicative identities $1_R$ and $1_S$. Let $I \trianglelefteq R$ be a two-sided ideal, and let $\varphi: R \to S$ be a unital ring homomorphism such that $I \subseteq \ker \varphi$, where $\ker \varphi=\{r \in R : \varphi(r)=0_S\}$. Then there exists a unique unital ring homomorphism

\begin{align*} \widetilde{\varphi}: R/I \to S \end{align*}

such that

\begin{align*} \widetilde{\varphi}(r+I) = \varphi(r) \end{align*}

for every $r \in R$.

Equivalently, if $\pi: R \to R/I$ is the quotient homomorphism defined by $\pi(r)=r+I$, then $\varphi = \widetilde{\varphi}\circ \pi$.