# 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 $$I\cap M=\emptyset$$.

Proofs: 1

Proofs: 1

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### References

#### Adapted from CC BY-SA 3.0 Sources:

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