◀ ▲ ▶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:

1. 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 \(I\).
2. 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 \(I\cap M=\emptyset\).
3. 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 BY-SA 4.0!

Github:

non-Github:

@Brenner

References

Adapted from CC BY-SA 3.0 Sources:

1. Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück