# 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.

#### 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