×

zbMATH — the first resource for mathematics

On the functionally countable subalgebra of \(C(X)\). (English) Zbl 1279.54015
For a completely regular Hausdorff space \(X\), let \(C(X)\) denote the ring of all real-valued continuous functions on \(X\). This paper deals with the subring \(C_c(X)\) of \(C(X)\) which contains all functions \(f\) in \(C(X)\) for which \(f(X)\) is countable. The authors observe that \(C_c(X)\) enjoys many properties analogous to \(C(X)\). An ideal \(I\) of \(C_c(X)\) is called a \(z_c\)-ideal if \(Z(f) \in Z_c[I]\), \(f\in C_c(X)\) \(\Rightarrow\) \(f\in I\) where \(Z_c[I] = \{Z(f) : f\in I\}\). \(z_c\)-ideals play a similar role as \(z\)-ideals in \(C(X)\).
The authors prove that (i) Every prime ideal in \(C_c(X)\) is contained in a unique maximal ideal in \(C_c(X)\). (ii) Every minimal prime ideal in \(C_c(X)\) is a \(z_c\)-ideal. (iii) Every prime ideal in \(C_c(X)\) is absolutely convex. (iv) If \(\{P_i\}_{i\in I}\) is a collection of semiprime ideals in \(C_c(X)\) such that \(P_i\) is a prime ideal for some \(i\in I\) then \(\sum_{i\in I} P_i\) is a prime ideal in \(C_c(X)\) or all of \(C_c(X)\). (v) If \(I\) is an ideal in \(C_c(X)\) then \(I\) and \(\surd I\) have the same largest \(z_c\)-ideal.
The authors show that for any space \(X\) (not necessarily completely regular), there is a zero-dimensional space \(Y\) which is a continuous image of \(X\) and \(C_c(X)\approx C_c(Y)\). It follows that for a space \(X\) with countably many components, there is a zero-dimensional space \(Y\) such that \(C_c(X)\approx C(Y)\). The authors call \(X\) a countably \(P\)-space (briefly, \(CP\)-space) if \(C_c(X)\) is a regular ring. It is shown that every \(P\)-space is a \(CP\)-space and every zero-dimensional \(CP\)-space is a \(P\)-space. Some topological and algebraic characterizations of \(CP\)-spaces are given which are parallel to the characterizations of \(P\)-spaces [L. Gillman and M. Jerison, Rings of continuous functions. Graduate Texts in Mathematics. 43. Springer-Verlag. (1976; Zbl 0327.46040)].
A commutative ring \(R\) is called self-injective (\(\aleph_0\)-selfinjective) if every homomorphism from an ideal (respectively countably generated ideal) of \(R\) into \(R\) can be extended to a homomorphism from \(R\) into \(R\). The authors prove that for a space \(X\), \(C_c(X)\) is a regular ring if and only if \(C_c(X)\) is an \(\aleph_0\)-selfinjective ring. Finally, an example of a space \(X\) is given for which \(C_c(X)\) is not isomorphic to any \(C(Y)\).

MSC:
54C40 Algebraic properties of function spaces in general topology
54C30 Real-valued functions in general topology
54C05 Continuous maps
54G10 \(P\)-spaces
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] F. AZARPANAH - O. A. S. KARAMZADEH - A. REZAEI ALIABAD, On z\?-ideals in C(X), Fund. Math. 160 (1999), pp. 15-25. · Zbl 0991.54014 · eudml:212377
[2] F. AZARPANAH - O. A. S. KARAMZADEH - S. RAHMATI, C(X) VS.C(X) Modulo its socle, Colloq. Math. 3 (2008), pp. 315-336. · Zbl 1149.54009 · doi:10.4064/cm111-2-9
[3] F. AZARPANAH - O. A. S. KARAMZADEH, Algebraic characterization of some disconnected spaces, Italian. J. Pure Appl. Math. 12 (2002), pp. 155-168. p^^^ ^^^^ p ^ · Zbl 1117.54030
[4] F. AZARPANAH - R. MOHAMADIAN, z-ideals and z\?-ideals in C(X), Acta. Math. Sin. (Engl. ser), 23 (2007), pp. 989-996. · Zbl 1186.54021 · doi:10.1007/s10114-005-0738-7
[5] T. DUBE, Contracting the Socle in Rings of Continuous Functions, Rend. Semin. Mat. Univ. Padova, 123 (2010), pp. 37-53. · Zbl 1200.06005 · doi:10.4171/RSMUP/123-2 · rendiconti.math.unipd.it · eudml:243271
[6] R. ENGELKING, General topology (Berlin: Heldermann, 1989).
[7] A. A. ESTAJI - O. A. S. KARAMZADEH, On C(X) Modulo its socle, Comm. Algebra, 31 (2003), pp. 1561-1571. · Zbl 1025.54012 · doi:10.1081/AGB-120018497
[8] M. GHADERMAZI - M. NAMDARI, On a-scattered spaces, Far East J. Math. Sci. (FJMS), 32 (2) (2009), pp. 267-274. · Zbl 1165.54310 · pphmj.com
[9] L. GILLMAN - M. JERISON, Rings of continuous functions (Springer, 1976). · Zbl 0093.30001
[10] K. R. GOODEARL, Von Neumann Regular Rings (Pitman, 1979).
[11] M. HENRIKSEN, Topology related to rings of real-valued continuous func- tions.Where it has been and where it might be going, Recent Progress In General Topology II, eds M. Husek and J. Van Mill (Elsevier Science, 2002), pp. 553-556.
[12] O. A. S. KARAMZADEH, On a question of Matlis, Comm. Algebra, 25 (1997), pp. 2717-2726. · Zbl 0878.16003 · doi:10.1080/00927879708826017
[13] O. A. S. KARAMZADEH, Modules whose countably generated submodules are epimorphic image, Colloq. Math. 2 (1982), pp. 7-10. · Zbl 0508.16024
[14] O. A. S. KARAMZADEH - A. A. KOOCHAKPOUR, On d -selfinjectivity of strongly 0 regular rings, Comm. Algebra, 27 (1999), pp. 1501-1513. · Zbl 0920.16001
[15] O. A. S. KARAMZADEH - M. MOTAMEDI - S. M. SHAHRTASH, On rings with a unique proper essential right ideal, Fund. Math. 183 (2004), pp. 229-244. · Zbl 1076.16017 · doi:10.4064/fm183-3-3
[16] O. A. S. KARAMZADEH - M. ROSTAMI, On the intrinsic topology and some related ideals of C(X), Proc. Amer. Math. Soc. 93 (1985), pp. 179-184. · Zbl 0524.54013 · doi:10.2307/2044578
[17] S. LARSON, A characterization of f -rings in which the sum of semiprime l- ideals is semiprime and its consequences, Comm. Algebra, (1995), pp. 5461-6481. · Zbl 0847.06007 · doi:10.1080/00927879508825545
[18] R. LEVY - M. MATVEEV, Functional separability, Comment. Math. Univ. Carolin. 51 (2010), pp. 705-711.
[19] R. LEVY - M. D. RICE, Normal P-spaces and the Gd-topology, Colloq. Math. 47 (1981), pp. 227-240.
[20] M. A. MULERO, Algebraic properties of rings of continuous functions, Fund. Math. 149 (1996), pp. 55-66. · Zbl 0840.54020 · eudml:212108
[21] A. PELCZYNSKI - Z . SEMADENI, Spaces of continuous functions (III), Studia Mathematica, 18 (1959), pp. 211-222.
[22] R. RAPHAEL - R. G. WOODS, On essential ring embeddings and the epimorphic hull of C(X), Theory and Applications of Categories, 14 (2005), pp. 46-52. · Zbl 1069.18001 · emis:journals/TAC/volumes/14/2/14-02abs.html · eudml:125005
[23] W. RUDIN, Continuous functions on compact spaces without perfect subsets, Proc. Amer. Math. Soc. 8 (1957), pp. 39-42. · Zbl 0077.31103 · doi:10.2307/2032807
[24] D. RUDD, On two sum theorems for ideals of C(X), Michigan Math. J. 17 (1970), pp. 139-141. · Zbl 0194.44403 · doi:10.1307/mmj/1029000423
[25] A. W. WICKSTEAD, An intrinsic Characterization of selfinjective semi-prime commutative rings, Proc. R. Ir. Acad. 90 (A) (1989), pp. 117-124. · Zbl 0677.13003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.