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Difference between revisions of "Structure space"

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''of a ring''
 
''of a ring''
  
The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090690/s0906901.png" /> of all its primitive ideals with the following topology: A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090690/s0906902.png" /> is closed if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090690/s0906903.png" /> contains every ideal that contains the intersection of all ideals from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090690/s0906904.png" /> (see [[Zariski topology|Zariski topology]]). The structure space of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090690/s0906905.png" /> is homeomorphic to the structure space of the quotient ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090690/s0906906.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090690/s0906907.png" /> is the Jacobson radical. A structure space is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090690/s0906908.png" />-space; if all primitive ideals of the ring are maximal, then the structure space is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090690/s0906909.png" />-space. The structure space of a ring with a unit is compact. The structure space of a biregular ring (see [[Regular ring (in the sense of von Neumann)|Regular ring (in the sense of von Neumann)]]) is locally compact and totally disconnected. It is used to represent a biregular ring in the form of a ring of continuous functions with compact supports.
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The set $\mathfrak P$ of all its primitive ideals with the following topology: A subset $C\subseteq\mathfrak P$ is closed if $C$ contains every ideal that contains the intersection of all ideals from $C$ (see [[Zariski topology|Zariski topology]]). The structure space of a ring $R$ is homeomorphic to the structure space of the quotient ring $R/J$, where $J$ is the Jacobson radical. A structure space is a $T_0$-space; if all primitive ideals of the ring are maximal, then the structure space is a $T_1$-space. The structure space of a ring with a unit is compact. The structure space of a biregular ring (see [[Regular ring (in the sense of von Neumann)|Regular ring (in the sense of von Neumann)]]) is locally compact and totally disconnected. It is used to represent a biregular ring in the form of a ring of continuous functions with compact supports.
  
 
====References====
 
====References====

Revision as of 09:52, 20 September 2014

of a ring

The set $\mathfrak P$ of all its primitive ideals with the following topology: A subset $C\subseteq\mathfrak P$ is closed if $C$ contains every ideal that contains the intersection of all ideals from $C$ (see Zariski topology). The structure space of a ring $R$ is homeomorphic to the structure space of the quotient ring $R/J$, where $J$ is the Jacobson radical. A structure space is a $T_0$-space; if all primitive ideals of the ring are maximal, then the structure space is a $T_1$-space. The structure space of a ring with a unit is compact. The structure space of a biregular ring (see Regular ring (in the sense of von Neumann)) is locally compact and totally disconnected. It is used to represent a biregular ring in the form of a ring of continuous functions with compact supports.

References

[1] N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956)


Comments

This is an extension of the notion of the spectrum space of maximal ideals of a commutative ring (cf. Spectrum of a ring).

References

[a1] K.R. Goodearl, "Von Neumann regular rings" , Pitman (1979)
How to Cite This Entry:
Structure space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Structure_space&oldid=33343
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article