\begin{definition}[Definition:Maximal Spectrum of Ring]
Let $A$ be a commutative ring with unity.
The '''maximal spectrum''' of $A$ is the set of maximal ideals of $A$:
:$\operatorname{Max} \: \Spec A = \set {\mathfrak m \lhd A : \mathfrak m \text { is maximal} }$
where $I \lhd A$ indicates that $I$ is an ideal of $A$.
The notation $\operatorname {Max} \: \Spec A$ is also a shorthand for the locally ringed space
:$\struct {\operatorname {Max} \: \Spec A, \tau, \OO_{\map {\operatorname {Max Spec} } A} }$
where:
:$\tau$ is the Zariski topology on $\map {\operatorname {Max Spec} } A$
:$\OO_{\map {\operatorname {Max Spec} } A}$ is the structure sheaf of $\map {\operatorname {Max Spec} } A$
\end{definition}