A general theory of structure spaces with applications to spaces of prime ideals

Abstract

A structure space is a quadrupleX=(X, d, A, P), where for some setR, X ⊂A=2^R,d:X×X →A is defined byd(I, J)=J−I, andP is the family of cofinite subsets ofR. Forr ε P, I ε X, N _r (I)={J ε X: d(I, J) ⊂r},To(X)={Q ⊂X: if x ε Q there is anr ε P such thatN _r (x)$\subseteq$Q}. ThenTo(X) is a (not usually Hausdorff) topology onX called the hull-kernel topology. Replacing d byd ^*, whered ^* (I, J)=d(J, I), or byd ^s, whered ^s (I, J.)=d(I, J) ∪d ^* (I, J), and proceeding in the obvious way yields thedual hull-kernel topology To(X ^*) andsymmetric topology To(X ^s). The latter is always a zero-dimensional Hausdorff space. When R is a commutative ring with identity andX is a collection of proper prime ideals ofR, To(X ^s) is usually called thepatch 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.