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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,...