Correspondences between ideals and z-filters for rings of continuous functions between C* and C

• Autorzy: Panman P. , Sack J. , Watson S.
• Języki publikacji: EN

Abstrakty

EN

Let X be a completely regular topological space. Let A(X) be a ring of continuous functions between C(X) and C{X), that is, C*{X) ⊆ A(X) ⊆ C(X). In [9], a correspondence Z_A between ideals of A(X) and z-filters on X is defined. Here we show that Z A extends the well-known correspondence for C* (X) to all rings A(X). We define a new correspondence 3 A and show that it extends the well-known correspondence for C(X) to all rings A{X). We give a formula that relates the two correspondences. We use properties of Z A. and 3 A to characterize C* (X) and C(X) among all rings A(X). We show that 3A defines a one-one correspondence between maximal ideals in A(X) and the z-ultrafilters in X.

Słowa kluczowe

EN

rings of continuous functions; ideals; z-filters; Kernels; hull

• Wydawca: Polskie Towarzystwo Matematyczne
• Czasopismo: Commentationes Mathematicae
• Rocznik: 2012
• Tom: Vol. 52, [Z] 1
• Strony: 11--20
• Opis fizyczny: Bibliogr. 11 poz.

Twórcy

autor

Panman P.

autor

Sack J.

autor

Watson S.

• California State University, Long Beach Department of Mathematics, California State University, Long Beach, Long Beach, CA 90840,

Bibliografia

• [1] W. Adamski; Two ultrafilter properties for vector lattices of real-valued functions, Publ. Math. Debrecen 45 (1994), 225-267.
• [2] H. L. Byun, and S. Watson; Prime and maximal ideals in subrings of C{X), Topology Appl. 40 (1991), 45-62.
• [3] H. L. Byun, and S. Watson; Local bounded inversion in rings of continuous functions, Comment. Math. 37 (1997), 55-63.
• [4] H. L. Byun, L. Redlin, and S. Watson; A-compactifications and rings of continuous functions between C* and C, Topological Vector Spaces, Algebras, and Related Areas, Pitman Research Notes in Mathematics Series 316 (1995), 130-139.
• [5] H. L. Byun, L. Redlin, and S. Watson; Local invertibility in subrings of C* (X)., Bull. Austral. Math. Soc. 46 (1992), no. 3, 449-458.
• [6] L. Gillman and M. Jerison; Rings of Continuous Functions, Springer-Verlag, New York, 1978.
• [7] M. Hendricksen, J. R. Isbell, and D. G. Johnson; Residue class fields of lattice-ordered algebras, Fund. Math. 50 (1961), 107-117.
• [8] D. Plank; On a class of subalgebras of C{X) with applications to (3X - X, Fund. Math. 64 (1969), 41-54.
• [9] L. Redlin and S. Watson; Maximal ideals in subalgebras of C(X), Proc. Amer. Math. Soc. 100 (1987), 763-766.
• [10] L. Redlin and S. Watson; Structure spaces for rings of continuous functions with applications to realcompactifications, Fund. Math. 152 (1997), 151-163.
• [11] S. Willard; General Topology, Addison-Wesley, Reading, MA, 1970.