#### Title

A General Theory of Structure Spaces with Applications to Spaces of Prime Ideals

#### Document Type

Article

#### Department

Mathematics (HMC)

#### Publication Date

9-1991

#### Abstract

A *structure space* is a quadruple *X = (X, d, A, P)*, where for some set *R*, *X c A = 2 ^{R}, d : X *×

*X*

*A*is defined by

*d(I, J) = J - I,*and

*P*is the family of cofinite subsets of

*R*. For

*r*ε

*P, I*ε

*X, N*ε

_{r}(I) = {J*X : d(I, J)*

*c*

*r}*,

*To(X) = {Q*

*c*

*X*: if x ε Q there is an

*r*ε

*P*such that

*N*}. Then

_{r}(x) ⊆ Q*To(X)*is a (not usually Hausdorfl) topology on

*X*called the

*hull-kernel topology*. Replacing

*d*by

*d**, where

*d*(I, J) = d(J, I),*or by

*d*where

^{s},*d*, and proceeding in the obvious way yields the

^{s}(I,J) = d(I, J) ∪ d*(I, J)*dual hull-kernel topology To(X*)*and

*symmetric topology To(X*The latter is always a zero-dimensional Hausdorff space. When

^{s}).*R*is a commutative ring with identity and

*X*is a collection of proper prime ideals of

*R*,

*To(X*is usually called the

^{s})*patch topology*. Our generality enables us to improve on known results in the case of space of prime ideals and to apply this theory to a wide variety of algebraic structures. In particular, we establish criteria for a subspace of a structure space to be closed in the symmetric topology; we establish a duality between families of maximal elements in the hull-kernel topology and families of minimal elements in the dual hull-kernel topology of subspaces that are closed in the symmetric topology; we use topological constructions to generalize certain ring theoretic notions, such as radical ideals an annihilator ideals; we use this theory to obtain new results about subspaces of the space prime ideals of a reduced, commutative ring.

#### Rights Information

© 1991 Birkhauser Verlag

#### Terms of Use & License Information

#### DOI

10.1007/BF01191086

#### Recommended Citation

Henriksen, M. and Kopperman, R. 1991. A general theory of structure spaces with applications to spaces of prime ideals. Algebra Universalis. 28(3):349-376.