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European Mathematical Society Publishing House
2016-09-19 17:05:49
Rendiconti del Seminario Matematico della Università di Padova
Rend. Sem. Mat. Univ. Padova
RSMUP
0041-8994
2240-2926
General
10.4171/RSMUP
http://www.ems-ph.org/doi/10.4171/RSMUP
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2013)
129
2013
0
On the Functionally Countable Subalgebra of $C(X)$
Mostafa
Ghadermazi
University of Kurdistan, SANANDAJ, IRAN
O.A.S.
Karamzadeh
Chamran University, AHVAZ, IRAN
M.
Namdari
Chamran University, AHVAZ, IRAN
Functionally countable subring, socle, $z_c$-idea, regular ring, $P$-space, $CP$-space, Scattered space, $\mathcal N_0$-selfinjective, zero-dimensional space, component
Let ${C_c{{\char 40}}X{{\char 41}}} =\{\,f\in C(X): f(X) \hbox{ is countable}\}$. Similar to $C(X)$ it is observed that the sum of any collection of semiprime (resp. prime) ideals in the ring ${C_c{{\char 40}}X{{\char 41}}} $ is either ${C_c{{\char 40}}X{{\char 41}}} $ or a semiprime (resp. prime) ideal in ${C_c{{\char 40}}X{{\char 41}}} $. For an ideal $I$ in ${C_c{{\char 40}}X{{\char 41}}} $, it is observed that $I$ and $\sqrt{I}$ have the same largest $z_c$-ideal. If $X$ is any topological space, we show that there is a zero-dimensional space $Y$ such that ${C_c{{\char 40}}X{{\char 41}}} \cong {C_c{{\char 40}}Y{{\char 41}}} $. Consequently, if $X$ has only countable number of components, then ${C_c{{\char 40}}X{{\char 41}}} \cong C(Y)$ for some zero-dimensional space $Y$. Spaces X for which ${C_c{{\char 40}}X{{\char 41}}} $ is regular (called $CP$-spaces) are characterized both algebraically and topo log ically and it is shown that $P$-spaces and $CP$-spaces coincide when $X$ is zero-dimensional. In contrast to $C^*(X)$, we observe that ${C_c{{\char 40}}X{{\char 41}}} $ enjoys the algebraic properties of regularity, $\aleph _{_0}$-selfinjectivity and some others, whenever $C(X)$ has these properties. Finally an example of a space $X$ such that ${C_c{{\char 40}}X{{\char 41}}} $ is not isomorphic to any $C(Y)$ is given.
General topology
Commutative rings and algebras
Associative rings and algebras
General
47
69
10.4171/RSMUP/129-4
http://www.ems-ph.org/doi/10.4171/RSMUP/129-4