Structure space
of a ring
The set of all its primitive ideals with the following topology: A subset
is closed if
contains every ideal that contains the intersection of all ideals from
(see Zariski topology). The structure space of a ring
is homeomorphic to the structure space of the quotient ring
, where
is the Jacobson radical. A structure space is a
-space; if all primitive ideals of the ring are maximal, then the structure space is a
-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) |
Structure space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Structure_space&oldid=33343