Let $R$ be a ring and $I$ be an ideal.
Is $R \setminus I = \{r \in R : r \notin I\}$ equal to the set $R - I =\{r- i : r \in R, i\in I\}$?
I think that they are the same but could not show whether it is true or false.
2026-04-06 17:47:38.1775497658
Is Ring\Ideal equal to Ring-Ideal as sets?
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No, this is false. $R\setminus I\subset R$ contains no elements of $I$, but as $0\in R\cap I$, clearly $0\in R-I$ since $0-0=0$. So the two are never equal as sets.