Atiyah-Macdonald before proposition 4.9

Question

Atiyah-Macdonald's proposition 4.9 is preceded by the following statement.

For any ideal $a$ and any multiplicatively closed subset $S$ in $A$, the contraction in $A$ of the ideal $S^{-1}a$ is denoted by $S(a)$.

I don't understand the meaning "$S(a)$".

If I consider $S(a)$ to be $\{sx\mid s\in S,x\in a\}$, then $S(a)$ becomes $a$. The notation seems odd.

Maybe I am not understanding correctly what contraction $(S^{-1}I)^c$ is.

How should this notation be considered? Please tell me.

Answer 1

It is right to say that $S(a)=(S^{-1}(a))^c$.
What is not true is that $(S^{-1}(a))^c=\{sx\mid s\in S,x\in a\}$.
The explicit expression of $(S^{-1}(a))^c$ is given in proposition 3.11, which states that $a^{ec}=\cup_{s\in S}(a:s)$. In general (as you can see also by the proposition you mention) the equality does not hold, i.e. $S(a)$ strictly contains $a$.

Answer 2

Given a ring homomorphism $f\colon A\to B$ and an ideal $I\subseteq B$, the contraction of $I$ (along $f$) is the preimage $f^{-1}(I)$, which is an ideal in $A$.

The definition of $S(\mathfrak{a})$ is the contraction of the ideal $S^{-1}\mathfrak{a}$ (along the canonical ring homomorphism $A\to S^{-1}A$).

Unpacking the definitions, we find that $x\in S(\mathfrak{a})$ if and only if $\frac{x}{1}=\frac{a}{s}$ for some $a\in \mathfrak{a}$ and $s\in S$. Equivalently, $x\in S(\mathfrak{a})$ if and only if there exists $s\in S$ such that $sx\in\mathfrak{a}$.