The space of minimal prime ideals of need not be basically disconnected

Authors:
A. Dow, M. Henriksen, Ralph Kopperman and J. Vermeer

Journal:
Proc. Amer. Math. Soc. **104** (1988), 317-320

MSC:
Primary 54C40; Secondary 06F25

DOI:
https://doi.org/10.1090/S0002-9939-1988-0958091-X

MathSciNet review:
958091

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Abstract: Problems posed twenty and twenty-five years ago by M. Henriksen and M. Jerison are solved by showing that the space of minimal prime ideals of the ring of continuous real-valued functions on a compact (Hausdorff) space need not be basically disconnected--or even an -space.

**[DHKV]**A. Dow, M. Henriksen, R. Kopperman and J. Vermeer,*The countable annihilator condition and weakly Lindelöf subspaces of minimal prime ideals*(submitted).**[GH]**L. Gillman and M. Henriksen,*Rings of continuous functions inn which finitely generated ideals are principal*, Trans. Amer. Math. Soc.**82**(1956), 366-391.**[GJ]**L. Gillman and M. Jerison,*Rings of continuous functions*, Van Nostrand, Princeton, N. J., 1960.**[HJ]**M. Henriksen and M. Jerison,*The space of minimal prime ideals of a commutative ring*, General Topology and its Relations to Modern Analysis and Algebra (Prague, 1962), and Trans. Amer. Math. Soc.**115**(1965), 110-130.**[Ho]**M. Hochster,*Prime ideal structure in commutative rings*, Trans. Amer. Math. Soc.**142**(1969), 43-60.**[K]**J. Kist,*Minimal prime ideals in commutative semigroups*, Proc. London Math. Soc. (3)**13**(1963), 31-50.**[Ku]**K. Kunen,*Some points in*, Math. Proc. Cambridge Philos. Soc.**80**(1976), 385-398.**[vM]**J. van Mill,*An introduction to*, Handbook of Set-Theoretic Topology, North-Holland, Amsterdam, 1984.**[W]**R. Walker,*The Stone-Čech compactification*, Springer-Verlag, Berlin and New York, 1974.**[Wo]**R. G. Woods,*A survey of absolutes of topological spaces*, Topological Structures II, Mathematical Centre Tracts, no. 116, 1979, pp. 323-362.

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DOI:
https://doi.org/10.1090/S0002-9939-1988-0958091-X

Article copyright:
© Copyright 1988
American Mathematical Society