Given a ring, or a -algebra (unital or not) , its maximal spectrum is the set of its maximal ideals.
If is a field, and is a finitely generated noetherian commutative unital -algebra without nilpotent elements, then equipped with the Zariski topology is a noetherian topological space; the varieties in the classical sense (cf. chapter 1 of Hartshorne) are exactly the spectra of such -algebras. A more appropriate spectrum for general commutative unital rings is the prime spectrum.
In functional analysis, there is a slight variant of this notion, defined using (automatically continuous) characters, the Gel'fand spectrum of a -algebra, where however the topology is much richer, indeed compact Hausdorff (locally compact Hausdorff, in the non-unital case).
In analytic geometry one also uses analytic spectra.