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"?
How comes that we have the "freshman algebraist's iso theorem"?
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abstract-algebra
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0$\frac{\frac{A}{C}}{\frac{B}{C}} = \frac{A}{C}\frac{C}{B}=\frac{A}{B}$ – 2012-01-14
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3So you're asking who first came up with this isomorphism theorem? They're often [attributed to Noether](http://en.wikipedia.org/wiki/Isomorphism_theorem#History), but see the article for details. – 2012-01-14
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0I 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...? – 2012-01-14
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1Oh, 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. – 2012-01-14
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1In 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