According to wikipedia, the saturation with respect to $f$ of an ideal $I$ in $R$ is the ideal $I:f^\infty:=\{g\in R:\exists k\in\mathbb{N}, f^kg\in I\}$.
The important property of the saturation, which ensures that it removes from the algebraic set defined by the ideal I the irreducible components on which the polynomial f is zero is the following: The primary decomposition of $I:f^\infty$ consists in the components of the primary decomposition of $I$ that do not contain any power of $f$.
Does anyone have references/books about the proof of this important property?
Given any set $S$ of polynomials in $R$, we denote by $V(S)$ the zero set, or variety of $S$: $V(S) = \{(p_1,\cdots,p_n) \in k^n | f(p_1,\cdots,p_n) = 0\ \text{for all } f \in S\}$. If $\mathfrak{a}=\langle S\rangle$, then $V(\mathfrak{a})=V(S)$. If $S=\{f\}$, then $V(S)=V(f)$.
According to Proposition 3.13 of the note "Grobner Bases: a Tutorial---Mike Stillman":
We have $V(I:f^\infty)=\overline{V(I)\backslash V(f)}$.
Does anyone have references/books contain a proof of this proposition?
Finally,
can one prove the important property by the Proposition 3.13?