Notation for ideal quotient

Question

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?

Answer 1

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 $\mathfrak a$.