View source for Structure space

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''of a ring $R$''

The set $\mathfrak P$ of all [[primitive ideal]]s $R$ 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====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  N. Jacobson,   "Structure of rings" , Amer. Math. Soc.  (1956)</TD></TR>
</table>



====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====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  K.R. Goodearl,   "Von Neumann regular rings" , Pitman  (1979)</TD></TR>
</table>