On ideals of ideals in $C(X)$

Document Type : Research Paper


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2 Chamran University‎, ‎Ahvaz‎,


In this article‎, ‎we have characterized ideals in $C(X)$ in which‎
‎every ideal is also an ideal (a $z$-ideal) of $C(X)$‎. ‎Motivated by‎
‎this characterization‎, ‎we observe that $C_infty(X)$ is a regular‎
‎ring if and only if every open locally compact $sigma$-compact‎
‎subset of $X$ is finite‎. ‎Concerning prime ideals‎, ‎it is shown that‎
‎the sum of every two prime (semiprime) ideals of each ideal in‎
‎$C(X)$ is prime (semiprime) if and only if $X$ is an $F$-space‎.
‎Concerning maximal ideals of an ideal‎, ‎we generalize the notion of‎
‎separability to ideals and we have proved the coincidence of‎
‎separability of an ideal with dense separability of a subspace of‎
‎$\beta X$‎. ‎Finally, we have shown that the Goldie dimension of an‎
‎ideal $I$ in $C(X)$ coincide with the cellularity of‎
‎$XsetminusDelta (I)$‎.


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