The Space of Minimal Prime Ideals of a Commutative Ring

Henriksen, Melvin; Jerison, M.

doi:10.2307/1994260

Cited by 44 publications

(69 citation statements)

References 3 publications

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“…However, as we saw in (2.3.12(i)) and the proof of (2.3.14), every infinite ultracopower via a countably incomplete ultrafilter contains a copy of /3(w). This space is known [22] …”

Section: Remark By (12) and The Initial Remarks Of ?21 F(z_x) Camentioning

confidence: 99%

Reduced coproducts of compact Hausdorff spaces

Bankston

1987

J. symb. log.

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By analyzing how one obtains the Stone space of the reduced product of an indexed collection of Boolean algebras from the Stone spaces of those algebras, we derive a topological construction, the “reduced coproduct”, which makes sense for indexed collections of arbitrary Tichonov spaces. When the filter in question is an ultrafilter, we show how the “ultracoproduct” can be obtained from the usual topological ultraproduct via a compactification process in the style of Wallman and Frink. We prove theorems dealing with the topological structure of reduced coproducts (especially ultracoproducts) and show in addition how one may use this construction to gain information about the category of compact Hausdorff spaces.

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“…However, as we saw in (2.3.12(i)) and the proof of (2.3.14), every infinite ultracopower via a countably incomplete ultrafilter contains a copy of /3(w). This space is known [22] …”

Section: Remark By (12) and The Initial Remarks Of ?21 F(z_x) Camentioning

confidence: 99%

Reduced coproducts of compact Hausdorff spaces

Bankston

1987

J. symb. log.

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“…We know / is a continuous bijection, and (a) is precisely the condition we need to make f~ι continuous. Now Min X is Hausdorff [4,Corollary 2.4] and Min X is extremal because it is dense in the extremal space X; on the other hand, r (f{X) is quasicompact because it is a quotient of the quasicompact space X, and since / is a homeomorphism, we obtain at once that both Min X and r έ? {X) are extremally disconnected Boolean spaces.…”

Section: E(y')) = φ(E(y))φ(e(y'))mentioning

confidence: 87%

Totally integrally closed rings and extremal spaces

Hochster

1970

Pacific J. Math.

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“…Unfortunately, KC(X) has no convenient dense subset (as does IX) and, before proceeding, we introduce the function 8X: Kc(X)->ßX defined by &x{P) -Pi where p is the unique [GJ,7.15] point in ßX with £ ç M". It is shown in [HJ,5.3] that 8X restricted to the minimal prime ideals topologized relative to Spec C{X) is continuous. The proof used there, however, readily adapts to show that 6X is continuous when the domain space is Spec C{X); since KC(X) is finer than Spec C{X), we have the continuity of 8X: KC(X) -> ßX.…”

Section: Px°fp=f°py-mentioning

confidence: 99%

“…The space m{A) of minimal prime ideals of A e 3ft topologized relative to Spec A has been studied extensively; see [KJ,[K2] and [HJ]. It is clear that the topology on the set m{A) relative to KA is finer than m{A); in fact, if A contains no nonzero nilpotent elements (in particular, if A = C{X)), then each A{a) n m{A) is open in m{A) [HJ,2.3] and these relative topologies coincide. We let m{X) denote the space m{C{X)) and note that m{X)<=tX [GJ,14.7].…”

Section: Corollary [Gj 14f]mentioning

confidence: 99%

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Structures determined by prime ideals of rings of functions

Montgomery¹

1970

Trans. Amer. Math. Soc.

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Introduction.Let 3t and S~z)s respectively denote the category of commutative rings with unity and the category of completely regular Hausdorff spaces; also, denotes the full-subcategory of ¡T^ whose objects are compact, and %>TD the full-subcategory of ^ whose objects are totally-disconnected. The collection of prime ideals of A e ¿% is the underlying set for an object KA e <ßTD and K is contravariantly functorial. If C denotes the contravariant functor which assigns to each Ie5¡¡ the ring C{X) of real-valued continuous functions on X, then the resulting functor Kc is the domain of a natural transformation 8: Kc-> ß, where ß denotes the Stone-Cech reflection of &~3i into &. The prime z-ideals of C{X) also furnish such a space IX, functor £ and natural transformation 8: £ -» ß. In the appropriate category, £ fills in a diagram which exhibits ß as a push-out.Topological properties of KC(X) and l,X are studied and IX is characterized as a certain compactification of XF, the £-topology on X, which helps establish the place of t,X between ßXP and ßX.The above results are applied in an investigation of the continuous and orderpreserved image of £ Y in both of ÇX and ßX arising from/: F-*-X. As one consequence, the prime z-ideal structure and the minimal prime ideal structure associated with X is illuminated by the corresponding structures associated with certain subspaces of X; as another, a convenient simplification and unification is provided for approaching several types of problems found in the literature on

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The Space of Minimal Prime Ideals of a Commutative Ring

Cited by 44 publications

References 3 publications

Reduced coproducts of compact Hausdorff spaces

Reduced coproducts of compact Hausdorff spaces

Totally integrally closed rings and extremal spaces

Structures determined by prime ideals of rings of functions

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