Abstract
It is shown thatX is finite if and only ifC(X) has a finite Goldie dimension. More generally we observe that the Goldie dimension ofC(X) is equal to the Souslin number ofX. Essential ideals inC(X) are characterized via their corresponding z-filters and a topological criterion is given for recognizing essential ideals inC(X). It is proved that the Fréchet z-filter (cofinite z-filter) is the intersection of essential z-filters. The intersection of idealsO x wherex runs through nonisolated points inX is the socle ofC(X) if and only if every open set containing all nonisolated points is cofinite. Finally it is shown that if every essential ideal inC(X) is a z-ideal thenX is a P-space.
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References
R. Engelking,General Topology, PWN polish scientific publishers, 1977.
N. J. Fine, L. Gillman andJ. Lambek,Rings of quotients of rings of functions, McGill Univ. Press, Montreal, 1966, MP # 635.
L. Gillman andM. Jerison,Rings of continuous functions, Springer-Verlag, 1976.
L. Gillman, Convex and Pseudo prime ideals inC(X), General topology and its applications, Proceedings of the 1988 Northeast Conference, New York (1990), 87–95.
K. R. Goodearl andR. B. Warfield, Jr.,An introduction to noncommutative Noetherian rings, Cambridge Univ. Press, 1989.
M. Henriksen andM. Jerison, The space of minimal prime ideals of commutative rings,Trans. Amer. Math. Soc. 115 (1965), 110–130.
R. E. Johnson, The extended centralizer of a ring over a module,Proc. Amer. Math. Soc. 2 (1951), 891–895.
I. Juhász, A. Verbeek andN. S. Kroonberg,Cardinal functions in topology, Math. Center Tracts,34, Amsterdam, 1971.
O. A. S. Karamzadeh, On the classical Krull dimension of rings,Fund. Math. 117 (1983), 103–108.
O. A. S. Karamzadeh andM. Rotami, On the intrinsic topology and some related ideals ofC(X).Proc. Amer. Math. Soc. 93(1), (1985), 179–184.
J. Lambek,Lectures on rings and modules, Blaisdell Publishing Company, 1966.
M. Mandelker, Supports of continuous functions,Trans. Amer. Math. Soc. 156 (1971), 73–83.
J. C. McConnel andJ. C. Robson,Noncommutative Noetherian rings, Wiley-Intersience, New York, 1987.
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Azarpanah, F. Essential ideals inC(X) . Period Math Hung 31, 105–112 (1995). https://doi.org/10.1007/BF01876485
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DOI: https://doi.org/10.1007/BF01876485