Structure space

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.

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

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