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|Paper IPM / M / 537||
An element of a ring R is called clean if it is the sum of a unit
and an idempotent and subset A of R is called clean if every
element of A is clean. A topological characterization of clean elements
of C(X) is given and it is shown that C(X) is clean if and only
if X is strongly zero-dimensional, if and only if there exists a
clean prime ideal in C(X). We will also characterize topological
space X for which the ideal CK(X) is clean.
Whenever X is locally compact, it is shown that
CK(X) is clean if and only if X is zero-dimensional.
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