I'm studying Ring Theory and I'm trying to develop an intuitive understanding of quotient rings. Am I justified in saying that $^{R}/_{I}$ is the collection of all subsets of $R$ whose union is the complement of $I$ in $R$?
2026-02-22 21:10:08.1771794608
Quotient rings as complements of ideals
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No, $R/I$ is the set of subsets of $R$ which are cosets of the group $(I,+)$ in $(R,+)$.
As such, they are pairwise disjoint and their union is all of $R$, not the complement of $I$ in $R$. One of them is, in fact, exactly $I$.