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.