Locally Socle of C(X)

Authors

1 Department of Mathematics, Shahid Chamran university

2 Department of Mathematics, Shahid Chamran University

Abstract

Let $LC_F(X)$ be the socle of $C(X)$ and $LC_F(X)={fin C(X) : overline{S_f}=X}$ , where $S_f$ is the union of all open subsets $U$ in $X$ such that $Ubackslash Z(f)|<infinity|$, $LC_F(X)$ is called the locally socle of $C(X)$ and it is a $z$-ideal of $C(X)$ containing $C_F(X)$. We characterize spaces $X$ for which the equality in the relation $C_F(X)subseteq LC_F(X)subseteq C(X)$ is hold. In fact, we show that $X$ is an almost discrete space if and only if $LC_F(X)=C(X)$. We note that if $X$ is an infinite space, then $C_F(X)subsetneq C(X)$. We also observe that $|I(X)|<infty$ if and only if $LC_F(X)=C_F(X)$. Moreover, it is shown that if $|I(X)|<infty$, then $LC_F(X)$ is never essential in any subring of $C(X)$ , while $LC_F(X)$ is an intersection of essential ideals of $C(X)$. We determine the conditions such that $LC_F(X)$ is not prime in any subring of $C(X)$ which contains the idempotents of $C(X)$. We investigate the primness of $LC_F(X)$ in some subrings of $C(X)$ .

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References

 Azarpanah, F. (1997). Intersection of essential ideals in C(X), Proc. Amer. Math. Soc.125, 2149-2154.
 Azarpanah, F. and Karamzadeh, O. A. S. (2002). Algebraic characterization of some disconnected spaces, Italian. J. Pure Appl. Math. 12, 155-168
 Azarpanah, F., Karamzadeh, O. A. S. and Rahmati, S. (2008). C(X) VS. C(X) modulo its socle, Coll. math. 3, 315-336.
 Estaji, A. and Karamzadeh, O. A. S. (2003). On C(X) Modulo its socle, Comm. Algebra 31, 1561-1571.
 Karamzadeh, O. A. S. and Rostami, M. (1985). On the intrinsic topology and some related ideals of C(X), Proc. Amer. Math. Soc. 93, 179-184.
 Henriksen, M. (2002).Topology related to rings of real-valued continuous functions. 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), pp: 553-556.
 Ghadermazi, M., Karamzadeh, O. A. S. and Namdari, M. (2013). On the functionally countable subalgebra of C(X), Rend. Sem. Mat. Univ. Padova, 129, 47-69.
 Ghadermazi, M., Karamzadeh, O. A. S. and Namdari, M. (2014). C(X) versus its functionally countable subalgebra, submitted to Fundamenta Mathematicae.
 Karamzadeh, O. A. S., Namdari, M. and Soltanpour, S. (2015). On the locally functionally countable subalgebra of C(X), Appl. Gen. Topol., 16, 183-207.
 Dube, T. (2010). Contracting the Socle in Rings of Continuous Functions, Rend. Semin. Mat. Univ. Padova. 123, 37-53.
 Engelking, R. (1989). General Topology, Heldermann Verlag Berlin.
 Gillman, L. and Jerison, M. (1976). Rings of continuous functions, Springer-Verlag.
 Goodearl, K. R. and Warfield R. B. (1989). An introduction to noncommutative noetherian rings, Cambridge University Press.
Journal of Advanced Mathematical Modeling
Volume 4, Issue 2 - Serial Number 2
October 2014
Pages 87-99

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History

  • Receive Date: 03 September 2014
  • Revise Date: 26 April 2015
  • Accept Date: 28 April 2015

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