Let $R$ be a unital ring and let $I \trianglelefteq R$ be a two-sided ideal. Let $R/I$ denote the set of additive cosets $a+I$ with $a \in R$. Define operations on $R/I$ by
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\begin{align*}
(a+I)+(b+I)=(a+b)+I
\end{align*}
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and
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\begin{align*}
(a+I)(b+I)=ab+I
\end{align*}
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for all $a,b \in R$. Then these operations are well-defined and make $R/I$ into a unital ring with additive identity $0_R+I$ and multiplicative identity $1_R+I$. The map
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\begin{align*}
\pi:R\to R/I
\end{align*}
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defined by $\pi(a)=a+I$ is a surjective unital ring homomorphism, and $\ker \pi=I$.