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Let $C_F(X)$ denote the socle of $C(X)$. It is shown that $X$ is a
$P$-space if and only if $C(X)$ is $\aleph_0$-selfinjective ring
or equivalently, if and only if $\frac{C(X)}{C_F(X)}$ is
$\aleph_0$-selfinjective. We also prove that $X$ is an extremally
disconnected $P$-space with only a finite number of isolated
points if and only if $\frac{C(X)}{C_F(X)}$ is selfinjective.
Consequently, if $X$ is a $P$-space, then $X$ is either an
extremally disconnected space with at most a countable number of
isolated points or both $C(X)$ and $\frac{C(X)}{C_F(X)}$ have
uncountable Goldie-dimensions. Prime ideals of
$\frac{C(X)}{C_F(X)}$ are also studied.
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