Got it. Thanks again Peter I appreciate it very much.

Thanks a lot Peter! The confusion indeed was caused by the omitting of the word proper in some places. Actually the interest in this terms aroused from trying to understsand the topology on the Stone-Cech compactification \beta(X) of a topological space X. For example, in your example of the space \beta(N), who would be the basic open sets for this topology?

Thanks. I was just wondering if a maximal ideal must contain the empty set..