Essential ideals in \(C(X)\). (English) Zbl 0869.54021

Azarpanah, F.

Period. Math. Hung. 31, No. 2, 105-112 (1995).

Summary: It is shown that \(X\) is finite if and only if \(C(X)\) has a finite Goldie dimension. More generally we observe that the Goldie dimension of \(C(X)\) is equal to the Souslin number of \(X\). Essential ideals in \(C(X)\) are characterized via their corresponding \(z\)-filters and a topological criterion is given for recognizing essential ideals in \(C(X)\). It is proved that the Fréchet \(z\)-filter (cofinite \(z\)-filter) is the intersection of essential \(z\)-filters. The intersection of ideals \(O_x\) where \(x\) runs through nonisolated points in \(X\) is the socle of \(C(X)\) if and only if every open set containing all nonisolated points is cofinite. Finally it is shown that if every essential ideal in \(C(X)\) is a \(z\)-ideal then \(X\) is a \(P\)-space.

MSC:

54C40	Algebraic properties of function spaces in general topology
13A18	Valuations and their generalizations for commutative rings

Keywords:

\(P\)-space; Fréchet filter; essential ideal; annihilator; Goldie dimension; Souslin number; socle

References:

[1]	R. Engelking,General Topology, PWN polish scientific publishers, 1977.
[2]	N. J. Fine, L. Gillman andJ. Lambek,Rings of quotients of rings of functions, McGill Univ. Press, Montreal, 1966, MP # 635. · Zbl 0143.35704
[3]	L. Gillman andM. Jerison,Rings of continuous functions, Springer-Verlag, 1976. · Zbl 0327.46040
[4]	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.
[5]	K. R. Goodearl andR. B. Warfield, Jr.,An introduction to noncommutative Noetherian rings, Cambridge Univ. Press, 1989. · Zbl 0679.16001
[6]	M. Henriksen andM. Jerison, The space of minimal prime ideals of commutative rings,Trans. Amer. Math. Soc. 115 (1965), 110–130. · Zbl 0147.29105 · doi:10.1090/S0002-9947-1965-0194880-9
[7]	R. E. Johnson, The extended centralizer of a ring over a module,Proc. Amer. Math. Soc. 2 (1951), 891–895. · Zbl 0044.02204 · doi:10.1090/S0002-9939-1951-0045695-9
[8]	I. Juhász, A. Verbeek andN. S. Kroonberg,Cardinal functions in topology, Math. Center Tracts,34, Amsterdam, 1971.
[9]	O. A. S. Karamzadeh, On the classical Krull dimension of rings,Fund. Math. 117 (1983), 103–108. · Zbl 0542.16022
[10]	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. · Zbl 0524.54013
[11]	J. Lambek,Lectures on rings and modules, Blaisdell Publishing Company, 1966. · Zbl 0143.26403
[12]	M. Mandelker, Supports of continuous functions,Trans. Amer. Math. Soc. 156 (1971), 73–83. · Zbl 0197.48703 · doi:10.1090/S0002-9947-1971-0275367-4
[13]	J. C. McConnel andJ. C. Robson,Noncommutative Noetherian rings, Wiley-Intersience, New York, 1987. · Zbl 0644.16008