# Understanding a lemma about minimal prime ideals (from McCoy's Rings and Ideals).

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.

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