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  • I think that if k is a ring and X is an infinite set, Spec(k^X) can be identified with the set of ultrafilters on X. The theorem that every ideal is contained in a maximal ideal implies the existence of non-principal ultrafilters (which correspond to maximal ideals of k^X other than the obvious ones), which logicians use to make constructions of non-standard models of the reals and the like.
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