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It should seem a silly (or even trivial) question, but I've always wondered who first noticed that quotient by an equivalence relation "really behaves like a fraction", meaning that (whatever this mean, depending on the setting you're working in) $ A/B \cong (A/C)/(B/C) $ everytime you have $C\le B\le A$. I mean, why on earth should I denote a set of equivalence classes by a relation -which I suppose just for the sake of simplicity to be a congruence on the set- with a fraction $\frac{\text{whole structure}}{\text{substructure}}$, if it wasn't for that useful "simplification"?

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    In the 1927 paper mentioned in the Wikipedia article she almost uses the modern notation: she writes $\mathfrak{R}\mid\mathfrak{a}$ for the quotient of a ring $\mathfrak{R}$ by an ideal $\mathfrak{a}$. She calls it a *Restklassenring*; this is literally 'residue class ring', and her *Restklasse* 'residue class' is an equivalence class under the congruence induced by the ideal.2012-01-14

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