Editing Spectrum of a ring (section)

== Examples ==
* The spectrum of integers: The affine scheme <math>\operatorname{Spec}(\mathbb{Z})</math> is the [[final object]] in the category of affine schemes since <math>\mathbb{Z}</math> is the [[initial object]] in the category of commutative rings.
* The scheme-theoretic analogue of <math>\mathbb{C}^n</math>: The affine scheme <math>\mathbb{A}^n_\mathbb{C} = \operatorname{Spec}(\mathbb{C}[x_1,\ldots, x_n])</math>. From the [[functor of points]] perspective, a point <math>(\alpha_1,\ldots,\alpha_n) \in \mathbb{C}^n</math> can be identified with the evaluation morphism <math>\mathbb{C}[x_1,\ldots, x_n]\xrightarrow[ev_{(\alpha_1,\dots,\alpha_n)}]{} \mathbb{C}</math>. This fundamental observation allows us to give meaning to other affine schemes.
* The cross: <math>\operatorname{Spec}(\mathbb{C}[x,y]/(xy))</math> looks topologically like the transverse intersection of two complex planes at a point (in particular, this scheme is not irreducible), although typically this is depicted as a <math>+</math>, since the only well defined morphisms to <math>\mathbb{C}</math> are the evaluation morphisms associated with the points <math>\{(\alpha_1,0), (0,\alpha_2) : \alpha_1,\alpha_2 \in \mathbb{C}  \}</math>.
* The prime spectrum of a [[Boolean ring]] (e.g., a [[power set ring]]) is a compact [[totally disconnected]] Hausdorff space (that is, a [[Stone space]]).{{sfnp|Atiyah|Macdonald|1969|loc=Ch. 1. Exercise 23. (iv)|ps=}}
* ([[Melvin Hochster|M. Hochster]]) A topological space is homeomorphic to the prime spectrum of a commutative ring (i.e., a [[spectral space]]) if and only if it is compact, [[quasi-separated space|quasi-separated]] and [[sober space|sober]].{{sfnp|Hochster|1969|ps=}}