Structure space

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of a ring \$R\$

The set \$\mathfrak P\$ of all primitive ideals \$R\$ 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=39351
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article