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I would like to know a little more about why we notate ideal quotients the way we do. Is the notation $(\mathfrak{a:b})$ supposed to connote the notion of ratio? Or is it better viewed as some sort of index, like the notation $|G:H|$ from group theory?

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In commutative algebra, the notation $(\mathfrak{a:b})$ is used for the set $\{x \in R \mid x\mathfrak{b} \subset \mathfrak a\}$, where $R$ is commutative ring and $\mathfrak {a, b}$ are ideals of $R$.

You can check that $(\mathfrak{a:b})$ is an ideal of $R$ (see Atiyah, MacDonald, Introduction to Commutative Algebra, 1969, p. 8). We have for instance $\mathfrak a \subset \mathfrak{(a:b)}$ (which means in some sense that $\mathfrak{(a:b)}$ divides $\mathfrak a$) and $(\mathfrak{a:b)b \subset a}$.

Let $R=\Bbb Z$. While $(\Bbb Z:2\Bbb Z)=\{x \in \Bbb Z \mid 2x\Bbb Z \subset \Bbb Z\} = \Bbb Z$ is infinite, the index of $2\Bbb Z$ in $\Bbb Z$ as additive groups is just $2$. So there is no direct relation between the notation $(\mathfrak{a:b})$ from ring theory and the notation $[G:H]$ from group theory. For instance, we don't even require $\mathfrak b$ to be a subset of $\mathfrak a$.

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    $\begingroup$ We also have the properties $((a:b):c)=((a:c):b)=(a:bc)$ (see Atiyah, MacDonald, Introduction to Commutative Algebra, 1969, p. 8), which translate the identities $$ \dfrac{ \frac{x}{y} }{z} = \dfrac{ \frac{x}{z} }{y} = \dfrac{ x }{yz}$$ from the real numbers $\endgroup$ – Watson Jan 16 '17 at 16:22
  • $\begingroup$ Moreover, $(0 : J) = \{x \in R \mid xJ = (0)\} = Ann_R(J)$ can be seen as the set of "zero-divisors w.r.t to $J$". If $J \subset I$ then $(I:J)=R$, because intuitively if a number $i$ divides $j$, say $j=n \cdot i$, then $\dfrac{i}{j}$ is the unit $\dfrac{1}{n}$ (and an ideal containing a unit is the whole ring). $\endgroup$ – Watson Jan 23 '17 at 12:50

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