Let $R$ be a ring with additive identity $0_R$, and let $I \trianglelefteq R$ be a two-sided ideal. For each $r \in R$, write $r+I := \{r+i : i \in I\}$ for the additive coset of $I$ in the additive group of $R$. Then for all $a,b \in R$,
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\begin{align*}
a+I=b+I \quad \iff \quad a-b \in I.
\end{align*}