nLab
maximal spectrum

Given a ring, or a $k$-algebra (unital or not) $A$, its maximal spectrum ${\mathrm{Spec}}_{m}A$ is the set of its maximal ideals.

If $k$ is a field, and $R$ is a finitely generated noetherian commutative unital $k$-algebra without nilpotent elements, then ${\mathrm{Spec}}_{m}A$ 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 $k$-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 continuous characters, the Gel'fand spectrum of a $C^{*}$-algebra, where however the topology is much richer, indeed (locally, in the non-unital case) compact Hausdorff.