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I am reading N. H. McCoy's book Rings and Ideals. In the section on minimal prime ideals belonging to (containing) an ideal, he proves the following lemma (lemma 3 in the section):

THEOREM A set $\mathfrak p$ of elements of the commutative ring $R$ is a minimal prime ideal belonging to the ideal $\mathfrak a$ if and only if $C(\mathfrak p)$ [complement of $\mathfrak p$] is a maximal multiplicative system which does not meet (is disjoint from) $\mathfrak a$.

We can order the set $S$ of all multiplicative systems which do not meet $\mathfrak a$ by inclusion. Then any chain in $S$ has an upper bound in $S$, since we can just take the (possibly infinite-fold) union of all the ideals in a chain $T$. By Zorn's maximality principle, we then get a unique maximal multiplicative system $M^*$ in $S$.

But then this would seem to imply that there can only be one minimal prime ideal belonging to any given ideal $\mathfrak a$. That would be $\mathfrak p^* = C(M^*)$. But this is a faulty conclusion and I do not know where I made a mistake. Any help would be appreciated.

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By Zorn's maximality principle, we then get a unique maximal multiplicative system $𝑀\ast$ in $𝑆$.

To me this suggests you are interpreting "maximal" as "maximum."

An element of a poset is a maximum if all other elements of the poset are less than it. A poset can have only one maximum element.

An element of a poset is maximal if there does not exist an element greater than it. A poset can have many maximal elements.

Zorn's conclusion does not say there is a maximum element, it says there exists a maximal element.

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    $\begingroup$ Oh! Thank you, in that case. Yes, you are right I did think that Zorn's lemma was supposed to postulate a maximum element. $\endgroup$
    – Jan Matula
    Commented Feb 19, 2023 at 16:32
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    $\begingroup$ @JanMatula You are not the first person to think so, either :) It's just one of those things... $\endgroup$
    – rschwieb
    Commented Feb 19, 2023 at 16:34

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