◀ ▲ ▶Branches / Algebra / Proposition: Open and Closed Subsets of a Zariski Topology
Proposition: Open and Closed Subsets of a Zariski Topology
For a commutative ring \(R\), the following properties are fulfilled:
 For an ideal \({I}\lhd R\), let \(R/I\) be the factor ring and let the function \(q\colon R\longrightarrow R/{I}\,\) have the corresponding spectrum function \(q^{ * }\colon \operatorname {Spec} \left(R/{I}\right)\longrightarrow \operatorname {Spec} \left(R\right)\,\). The image of the spectrum function \(q^{ * }\) is closed. It is the set \(V(I)\) of all ideals of \(R\) containing \(I\).
 For a multiplicative system \(M\subseteq R\) let the canonical function \(\iota \colon R\longrightarrow R_{M}\,\) have the corresponding spectrum function \(\iota ^{ * }\colon \operatorname {Spec} \left(R_{M}\right)\longrightarrow \operatorname {Spec} \left(R\right)\,\). The spectrum function \(\iota ^{ * }\) is injective and the image consists of all prime ideals \(I\lhd M\) with \(I\cap M=\emptyset\).
 For a multiplicative system \(M\subseteq R\) let the canonical function \(\iota \colon R\longrightarrow R_{M}\,\) have the corresponding spectrum function \(\iota ^{ * }\colon \operatorname {Spec} \left(R_{M}\right)\longrightarrow \operatorname {Spec} \left(R\right)\,\). The spectrum function \(\iota ^{ * }\) is injective and the image consists of all prime ideals \(I\lhd M\) with \(I\cap M=\emptyset\).
Table of Contents
Proofs: 1
Mentioned in:
Proofs: 1
Thank you to the contributors under CC BYSA 4.0!
 Github:

 nonGithub:
 @Brenner
References
Adapted from CC BYSA 3.0 Sources:
 Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of OsnabrÃ¼ck