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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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$\frac{\frac{A}{C}}{\frac{B}{C}} = \frac{A}{C}\frac{C}{B}=\frac{A}{B}$ –  tetrapharmakon Jan 14 '12 at 18:20
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So you're asking who first came up with this isomorphism theorem? They're often attributed to Noether, but see the article for details. –  Dylan Moreland Jan 14 '12 at 18:22
    
I found the article but I don't understand any German. So you're saying that Noether first used the fractional notation. But what was her motivation? Did she find the iso theorem and then defined quotients, or...? –  tetrapharmakon Jan 14 '12 at 18:30
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Oh, I meant the Wikipedia article. I don't read German either, and I don't know anything about her motivation or notation. I only hoped that some bit of historical information might get you started. –  Dylan Moreland Jan 14 '12 at 18:35
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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. –  Brian M. Scott Jan 14 '12 at 20:21

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