nLab
maximal spectrum

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Definition

Given a ring, or a kk-algebra (unital or not) AA, its maximal spectrum Spec mASpec_m A is the set of its maximal ideals.

Properties

If kk is a field, and RR is a finitely generated noetherian commutative unital kk-algebra without nilpotent elements, then Spec mASpec_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 kk-algebras. A more appropriate spectrum for general commutative unital rings is the prime spectrum.

Revised on January 6, 2012 10:57:17 by Urs Schreiber (89.204.137.240)