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

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Mathematics (HMC)

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A structure space is a quadruple X = (X, d, A, P), where for some set R, X c A = 2R, 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, Nr(I) = {J ε X : d(I, J) c r}, To(X) = {Q c X : if x ε Q there is an r ε P such that Nr(x) ⊆ Q}. Then 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 ds, where ds(I,J) = d(I, J) ∪ d*(I, J), and proceeding in the obvious way yields the dual hull-kernel topology To(X*) and symmetric topology To(Xs). The latter is always a zero-dimensional Hausdorff space. When R is a commutative ring with identity and X is a collection of proper prime ideals of R, To(Xs) is usually called the 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.

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© 1991 Birkhauser Verlag

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