Difference between revisions of "Noetherian space"
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| − | Examples of Noetherian spaces are some spectra of commutative rings (cf. [[ | + | A [[topological space]] $X$ which satisfies the [[descending chain condition]] for closed subspaces: every strictly decreasing chain of closed subspaces breaks off. An equivalent condition is: Any non-empty family of closed subsets ordered by inclusion has a minimal element. Every subspace of a Noetherian space is itself Noetherian. If a space $X$ has a finite covering by Noetherian subspaces, then $X$ is itself Noetherian. A space $X$ is Noetherian if and only if every open subset of $X$ is quasi-compact. A Noetherian space is the union of finitely many irreducible components. |
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| + | Examples of Noetherian spaces are some spectra of commutative rings (cf. [[Spectrum of a ring]]). For a ring $A$ the space $\mathrm{Spec}(A)$ (the spectrum of $A$) is Noetherian if and only if $A/J$ is a [[Noetherian ring|Noetherian ring]], where $J$ is the [[nil radical]] of $A$. | ||
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) {{MR|0360549}} {{ZBL|0279.13001}} </TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) {{MR|0360549}} {{ZBL|0279.13001}} </TD></TR> | ||
| + | </table> | ||
Latest revision as of 17:12, 29 December 2020
2020 Mathematics Subject Classification: Primary: 54F65 [MSN][ZBL]
A topological space $X$ which satisfies the descending chain condition for closed subspaces: every strictly decreasing chain of closed subspaces breaks off. An equivalent condition is: Any non-empty family of closed subsets ordered by inclusion has a minimal element. Every subspace of a Noetherian space is itself Noetherian. If a space $X$ has a finite covering by Noetherian subspaces, then $X$ is itself Noetherian. A space $X$ is Noetherian if and only if every open subset of $X$ is quasi-compact. A Noetherian space is the union of finitely many irreducible components.
Examples of Noetherian spaces are some spectra of commutative rings (cf. Spectrum of a ring). For a ring $A$ the space $\mathrm{Spec}(A)$ (the spectrum of $A$) is Noetherian if and only if $A/J$ is a Noetherian ring, where $J$ is the nil radical of $A$.
References
| [1] | N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) MR0360549 Zbl 0279.13001 |
Comments
References
| [a1] | R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001 |
How to Cite This Entry:
Noetherian space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Noetherian_space&oldid=23912
Noetherian space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Noetherian_space&oldid=23912
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article