Definition:Maximal Spectrum of Ring

Definition
Let $A$ be a commutative ring with unity.

The maximal spectrum of $A$ is the set of maximal ideals of $A$:


 * $\operatorname{Max}\:\operatorname{Spec} \left({A}\right) = \left\{{\mathfrak m \lhd A : \mathfrak m \text{ is maximal}}\right\}$

where $I \lhd A$ indicates that $I$ is an ideal of $A$.

The notation $\operatorname{Max}\:\operatorname{Spec} \left({A}\right)$ is also a shorthand for the locally ringed space
 * $(\operatorname{Max}\:\operatorname{Spec} \left({A}\right), \tau, \mathcal O_{\operatorname{MaxSpec}(A)})$

where:
 * $\tau$ is the Zariski topology on $\operatorname{MaxSpec}(A)$
 * $\mathcal O_{\operatorname{MaxSpec}(A)}$ is the structure sheaf of $\operatorname{MaxSpec}(A)$

The maximal spectrum of a ring is understood to be a topological space: it inherits the subspace topology from the prime spectrum.

Also see

 * Definition:Prime Spectrum of Ring
 * Definition:Affine Algebraic Variety