# Contents

## Definition

Given a ring, or a $k$-algebra (unital or not) $A$, its maximal spectrum $Spec_m A$ is the set of its maximal ideals.

## Properties

If $k$ is a field, and $R$ is a finitely generated noetherian commutative unital $k$-algebra without nilpotent elements, then $Spec_m A$ 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.

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