“School of Mathematics”
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| Paper IPM / M / 492 |
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| Abstract: | |||||
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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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