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.
Best Answer
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 $a$.