I'm trying to understand a particular proof, but it has a step which isn't really given any justification. Here is the part that's giving me trouble:
"Let be a semiprime ideal of a semigroup , and let . Letting , we obtain an m-system disjoint from ."
For those unfamiliar with the terminology, a semiprime ideal is an ideal of a semigroup with the property that for any . An m-system is a non-empty subset of S such that there exists such that .
What I'm trying to understand is why should be disjoint from . I understand why it is an m-system, but I don't understand why, say, can't be in . Any insights would be appreciated.