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An element $f$ of a commutative ring $A$ with identity element is called a if there is a $g$ in $A$ such that $f^{2}g=f$. A point $p$ of a (Tychonoff) space $X$ is called a $P$- if each $f$ in the ring $C\left(X\right)$ of continuous real-valued functions is constant on a neighborhood of $p$. It is well-known that the ring $C\left(X\right)$ is von Neumann regular ring iff each of its elements is a von Neumann regular element; in which case $X$ is called a $P$-. If all but at most one point of $X$ is a $P$-point, then $X$ is called an . In earlier work it was shown...

In 1950 in volume 1 of Proc. Amer. Math. Soc., B. Brown and N. McCoy showed that every (not necessarily commutative) ring $R$ has an ideal $𝔐\left(R\right)$ consisting of elements $a$ for which there is an $x$ such that $axa=a$, and maximal with respect to this property. Considering only the case when $R$ is commutative and has an identity element, it is often not easy to determine when $𝔐\left(R\right)$ is not just the zero ideal. We determine when this happens in a number of cases: Namely when at least one of $a$ or $1-a$ has a von Neumann inverse,...