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\(C(X)\) Versus its Functionally Countable Subalgebra

  • Mostafa Ghadermazi
  • Omid Ali S. Karamzadeh
  • Mehrad Namdari
Original Paper
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Abstract

Let \(C_c(X)\) (resp. \(C^F(X)\)) denote the subring of \(C(X)\) consisting of functions with countable (resp. finite) image and \(C_F(X)\) be the socle of \(C(X)\). We characterize spaces X with \(C^*(X)=C_c(X)\), which generalizes a celebrated result due to Rudin, Pelczynnski, and Semadeni. Two zero-dimensional compact spaces X, Y are homeomorphic if and only if \(C_c(X)\cong C_c(Y)\) (resp. \(C^F(X)\cong \ C^F(Y)\)). The spaces X for which \(C_c(X)=C^F(X)\) are characterized. The socles of \(C_c(X)\), \(C^F(X)\), which are observed to be the same, are topologically characterized and spaces X for which this socle coincides with \(C_F(X)\) are determined, too. A certain well-known algebraic property of \(C(X)\), where X is real compact, is extended to \(C_c(X)\). In contrast to the fact that \(C_F(X)\) is never prime in \(C(X)\), we characterize spaces X for which \(C_F(X)\) is a prime ideal in \(C_c(X)\). It is observed for these spaces, \(C_c(X)\) coincides with its own socle (a fact, which is never true for \(C(X)\), unless X is finite). Finally, we show that a space X is the one-point compactification of a discrete space if and only if \(C_F(X)\) is a unique proper essential ideal in \(C^F(X)\).

Keywords

Functionally countable subring Socle \(z_c\)-ideal Regular ring CP-space 

Mathematics Subject Classification

Primary 54C40 Secondary 13C11 

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Copyright information

© Iranian Mathematical Society 2018

Authors and Affiliations

  • Mostafa Ghadermazi
    • 1
  • Omid Ali S. Karamzadeh
    • 2
  • Mehrad Namdari
    • 2
  1. 1.Department of MathematicsUniversity of KurdistanSanandajIran
  2. 2.Department of MathematicsShahid Chamran University of AhvazAhvazIran

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