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Paper   IPM / M / 492
School of Mathematics
  Title:   Algebraic characterization of some disconnected spaces
  Author(s): 
1.  O. A. S. Karamzadeh
2.  F. Azarpanah
  Status:   Published
  Journal: Ital. J. pure Appl. Math.
  No.:  1
  Year:  2002
  Pages:   155-168
  Supported by:  IPM
  Abstract:
It is shown that X is an extremally disconnected space if and only if C(x) is a Bear ring. We also give several new algebraic characterizations of basically disconnected spaces. These characterizations are then used to give a unified proof of the fact that X is extremally (basically) disconnected space if and only if βX is extremally (basically) disconnected space. Zero-dimensional and strongly zero-dimensional spaces are also characterized similarly. It is shown that X is strongly zero-dimensional F-space if and only if each minimal prime ideal in C(X) is generated by idempotents. We also show that X is an extremally disconnected P-space with a dense set of isolated points if and only if C(X) is isomorphic to a direct product of fields. Finally, we prove that C(X) is a self injective ring if and only if X is an extremally disconnected P-space.

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