### On the locally functionally countable subalgebra of C(X) on locally functionally countable subalgebra of C(X)

#### Abstract

Let $C_c(X)=\{f\in C(X) : |f(X)|\leq \aleph_0\}$, $C^F(X)=\{f\in C(X): |f(X)|<\infty\}$, and $L_c(X)=\{f\in C(X) : \overline{C_f}=X\}$, where $C_f$ is the union of all open subsets $U\subseteq X$ such that $|f(U)|\leq\aleph_0$, and $C_F(X)$ be the socle of $C(X)$ (i.e., the sum of minimal ideals of $C(X)$). It is shown that if $X$ is a locally compact space, then $L_c(X)=C(X)$ if and only if $X$ is locally scattered.

We observe that $L_c(X)$ enjoys most of the important properties which are shared by $C(X)$ and $C_c(X)$.

Spaces $X$ such that $L_c(X)$ is regular (von Neumann) are characterized. Similarly to $C(X)$ and $C_c(X)$, it is shown that $L_c(X)$ is a regular ring if and only if it is $\aleph_0$-selfinjective.

We also determine spaces $X$ such that ${\rm Soc}{\big(}L_c(X){\big)}=C_F(X)$ (resp., ${\rm Soc}{\big(}L_c(X){\big)}={\rm Soc}{\big(}C_c(X){\big)}$). It is proved that if $C_F(X)$ is a maximal ideal in $L_c(X)$, then $C_c(X)=C^F(X)=L_c(X)\cong \prod\limits_{i=1}^n R_i$, where $R_i=\mathbb R$ for each $i$, and $X$ has a unique infinite clopen connected subset. The converse of the latter result is also given. The spaces $X$ for which $C_F(X)$ is a prime ideal in $L_c(X)$ are characterized and consequently for these spaces, we infer that $L_c(X)$ can not be isomorphic to any $C(Y)$.

#### Keywords

#### Full Text:

PDF#### References

. Azarpanah, Intersection of essential ideals in C(X), Proc. Amer. Math. Soc. 125 (1997), 2149-2154.

http://dx.doi.org/10.1090/S0002-9939-97-04086-0

F. Azarpanah and O. A. S. Karamzadeh, Algebraic characterization of some disconnected spaces, Italian. J. Pure Appl. Math. 12 (2002), 155-168.

F. Azarpanah, O. A. S. Karamzadeh and S. Rahmati, C(X) vs. C(X) modulo its socle, Colloq. Math. 3 (2008),315-336.

http://dx.doi.org/10.4064/cm111-2-9

P. Bhattacharjee, M. L. Knox and W. Wm. Mcgovern, The classical ring of quotients of $C_c(X)$, Appl. Gen. Topol.15, no. 2 (2014), 147-154.

http://dx.doi.org/10.4995/agt.2014.3181

O. Dovgoshey, O.Martio, V. Ryazanov and M. Vuorinen, The Cantor function, Expo. Math. 24 (2006), 1-37.

http://dx.doi.org/10.1016/j.exmath.2005.05.002

T. Dube, Contracting the Socle in Rings of Continuous Functions, Rend. Semin. Mat. Univ. Padova. 123 (2010), 37-53.

http://dx.doi.org/10.4171/RSMUP/123-2

R. Engelking, General Topology, Heldermann Verlag Berlin, 1989.

A. A. Estaji and O. A. S. Karamzadeh, On C(X) modulo its socle, Comm. Algebra 31 (2003), 1561-1571.

http://dx.doi.org/10.1081/AGB-120018497

M. Ghadermazi, O. A. S. Karamzadeh and M. Namdari, On the functionally countable subalgebra of $C(X)$, Rend. Sem. Mat. Univ. Padova, 129 (2013), 47-69.

http://dx.doi.org/10.4171/RSMUP/129-4

S. G. Ghasemzadeh, O. A. S. Karamzadeh and M. Namdari, The super socle of the ring of continuous functions, Mathematica Slovaka, to appear.

J. Hart and K. Kunen, Locally constant functions, Fund. Math. 150 (1996), 67-96.

M. Henriksen, R. Raphael and R. G. Woods, SP-scattered spaces; a new generalization of scattered spaces, Comment. Math. Univ. Carolin 48, no. 3 (2007), 487-505.

O. A. S. Karamzadeh, On a question of Matlis, Comm. Algebra 25 (1997), 2717-2726.

http://dx.doi.org/10.1080/00927879708826017

O. A. S. Karamzadeh and A. A. Koochakpour, On $aleph_{_0}$-selfinjectivity of strongly regular rings, Comm. Algebra 27 (1999), 1501-1513.

http://dx.doi.org/10.1080/00927879908826510

O. A. S. Karamzadeh, M. Namdari and M. A. Siavoshi, A note on $lambda$-compact spaces, Math. Slovaca. 63, no. 6 (2013) 1371-1380.

http://dx.doi.org/10.2478/s12175-013-0177-3

O. A. S. Karamzadeh and M. Rostami, On the intrinsic topology and some related ideals of C(X), Proc. Amer. Math. Soc. 93 (1985), 179-184.

R. Levy and M. D. Rice, Normal P-spaces and the $G_delta$-topology, Colloq. Math. 47 (1981), 227-240.

M. A. Mulero, Algebraic properties of rings of continuous functions, Fund. Math. 149 (1996), 55-66.

M. Namdari and A. Veisi, The subalgebra of $C_c(X)$ consisting of elements with countable image versus C(X) with respect to their rings of quotients, Far East J. Math. Sci. (FJMS), 59 (2011), 201-212.

M. Namdari and A. Veisi, Rings of quotients of the subalgebra of C(X) consisting of functions with countable image, Inter. Math. Forum, 7 (2012), 561-571.

A. Pelczynski and Z. Semadeni, Spaces of continuous functions (III), Studia Mathematica 18 (1959), 211-222.

M. E. Rudin and W. Rudin, Continuous functions that are locally constant on dense sets, J. Funct. Anal. 133 (1995), 120-137.

http://dx.doi.org/10.1006/jfan.1995.1121

W. Rudin, Continuous functions on compact spaces without perfect subsets, Proc. Amer. Math. Soc. 8 (1957), 39-42.

http://dx.doi.org/10.1090/S0002-9939-1957-0085475-7

D. Rudd, On two sum theorems for ideals of C(X), Michigan Math. J. 17 (1970), 139-141.

http://dx.doi.org/10.1307/mmj/1029000423

Abstract Views

_{Metrics powered by PLOS ALM}

**Cited-By (articles included in Crossref)**

This journal is a Crossref Cited-by Linking member. This list shows the references that citing the article automatically, if there are. For more information about the system please visit Crossref site

1. **Ordered Field Valued Continuous Functions with Countable Range**

Sudip Kumar Acharyya, Atasi Deb Ray, Pratip Nandi** Bulletin of the Iranian Mathematical Society** year: 2021

doi: 10.1007/s41980-021-00540-8

Esta revista se publica bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional. Universitat Politècnica de València e-ISSN: 1989-4147 https://doi.org/10.4995/agt |