An element of a ring R is called clean if it is the sum of a unit and an idempotent and a 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 spaces 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.
Unable to display preview. Download preview PDF.
- V. P. Camillo and H. P. Yu, Exchange rings, units and idempotents, Comm. Alg., 22 (1994), 4737–4749.Google Scholar
- R. Engelking, General Topology, PWN Polish Scientific Publishers (1977).Google Scholar
- L. Gillman and M. Jerison, Rings of Continuous Functions, Springer-Verlag (1976).Google Scholar
- O. A. S. Karamzadeh and M. Rostami, On the intrinsic topology and some related ideals of C(X), Proc. Amer. Math. Soc., 93 (1985), 179–184.Google Scholar
- G. S. Monk, A characterization of exchange rings, Proc. Amer. Math. Soc., 35 (1972), 344–353.Google Scholar
- W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc., 229 (1977), 269–278.Google Scholar
- R. B. Warfield, A Krull-Schmidt theorem for infinite sums of modules, Proc. Amer. Math. Soc., 22 (1969), 460–465.Google Scholar