TITLE: Characterization of the saturation of a multiplicative set
QUESTION [2 upvotes]: Recently, I proved the following nice characterization of the saturation of the multiplicative subset $ S_f: = \{1,f,f^2,\ldots\}$ for $f$ an element of a commutative unital ring $A$ (however, I only need to work with $A$ a f.g. $k$-algebra over a field $k$). 
Let me denote by $\overline{S}$ to the saturation of a multiplicative set $S$. I proved that,
$$\overline{S_f}=A-\bigcup\{ m \mid m \text{ maximal and does not contain } f\}$$
My problem is this result will automatically imply the following —unexpected— equality.
$$\bigcup\{ m \mid m \text{ maximal ideal in A that does not contain } f\} = \bigcup\{ p \mid p \text{ prime ideal in A that does not contain } f\}$$
Which I simply do not believe. But I tried to produce a counter-example using schemes and it looks the examples are avoiding the difference between those two sets.

REPLY [4 votes]: Consider the maximal ideal $5\mathbb{Z}$ of the ring $(\mathbb{Z}, +, .)$. Then for the ring $\mathbb{Z}_{5\mathbb{Z}}$, the localization of $\mathbb{Z}$ at $5\mathbb{Z}$, we have $Spec(\mathbb{Z}_{5\mathbb{Z}}=\{0, 5\mathbb{Z}_{5\mathbb{Z}}\}$. Now set $f=5/1(\in \mathbb{Z}_{5\mathbb{Z}})$. Thus we have 
$$\bigcup\{ m \mid m \text{ maximal ideal in $\mathbb{Z}_{5\mathbb{Z}}$ that does not contain } f\} =\{5\mathbb{Z}_{5\mathbb{Z}}\},$$ and$$ \bigcup\{ p \mid p \text{ prime ideal in $\mathbb{Z}_{5\mathbb{Z}}$ that does not contain } f\}=\{0\}.$$
Hence, 
$$\mathbb{Z}_{5\mathbb{Z}}-\bigcup\{ m \mid m \text{ maximal and does not contain } f\}\ne \mathbb{Z}_{5\mathbb{Z}}-\bigcup\{ p \mid p \text{ prime and does not contain } f\}.$$
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