Let $Sigma (X,mathbb{C})$ denote the collection of all the rings between $C^*(X,mathbb{C})$ and $C(X,mathbb{C})$. We show that there is a natural correlation between the absolutely convex ideals/ prime ideals/maximal ideals/$z$-ideals/$z^circ$-ideals in the rings $P(X,mathbb{C})$ in $Sigma(X,mathbb{C})$ and in their real-valued counterparts $P(X,mathbb{C})cap C(X)$. It is shown that the structure space of any such $P(X,mathbb{C})$ is $beta X$. We show that for any maximal ideal $M$ in $C(X,mathbb{C}), C(X,mathbb{C})/M$ is an algebraically closed field. We give a necessary and sufficient condition for the ideal $C_{mathcal{P}}(X,mathbb{C})$ of $C(X,mathbb{C})$ to be a prime ideal, and we examine a few special cases thereafter.
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