Atiyah Macdonald chapter 1, question 21 iii

Asked Sep 09 '19 at 13:05

Active Sep 09 '19 at 16:51

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In a commutative ring $A$ with $1$ having a subset $S$ and some element $x \notin S$, does there always exist a prime ideal containing $S$ but not $x$? I am trying to solve the $\Rightarrow$ part of (iii). If this is not true always,can you help me figure out a solution? Thanks in advance.

(Edit) I could not understand the solution given in the link.

edited Sep 09 '19 at 16:51

user26857

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asked Sep 09 '19 at 13:05

jimm

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But that still doesn't answer my initial question. – jimm Sep 09 '19 at 13:17
True. But I fear you might be having an XY problem. Consider $A = \Bbb Z$ and $S = {4, 6, 8}$ and $x = 2$, for instance. – Arthur Sep 09 '19 at 13:21
Ok. The thing is I have seen the solution also, but didn't understand. So can you please tell me the solution in a simpler language? – jimm Sep 09 '19 at 13:29

Atiyah Macdonald chapter 1, question 21 iii

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