On regular ideals in reduced rings

Let $R$ be a commutative ring with identity and $X$ be a Tychonoff space. An ideal $I$ of $R$ is Von Neumann regular (briefly, regular) if for every $a\in I$, there exists $b\in R$ such that $a=a^2b$. In the present paper, we obtain the general form of a regular ideal in $C(X)$ which is $O^A$, for some closed subset $A$ of ${\beta X}$, for which $ A^c\cap X\subseteq (P(X))^{\circ }$, where $P(X)$ is the set of all $P$-points of $X$. We show that the ideals and subrings such as $C_K(X)$, $C_\psi(X)$, $C_\infty(X)$, $Soc_mC(X)$ and $M^{\beta X\backslash X}$ are regular if and only if they are equal to the socle of $C(X)$. We carry further the study of maximal regular ideal, for instance, it is shown that for a vast class of topological spaces (we call them $OPD$-spaces) the maximal regular ideal is $O^{X\backslash I(X)}$, where $I(X)$ is the set of nonisolated points of $X$. Also, for this class, the socle of $C(X)$ is the maximal regular ideal if and only if $I(X)$ contains no infinite closed set. We also show that $C(X)$ contains an ideal which is both essential and regular if and only if $(P(X))^{\circ }$ is dense in $X$. Finally it is shown that, for semiprimitive rings pure ideals are of the form $O^A$ which $A$ is a closed subset of $Max(R)$, also a $P$-point of $X=Max(R)$ is introduced and it is shown that the maximal regular ideal of an arbitrary ring $R$ is $O^{X\backslash P(X)}$, which $P(X)$ is the set of $P$-points of $X=Max(R)$