The maximal regular ideal of some commutative rings

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 (sic)(R) 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 (sic)(R) 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, when R is a product of local rings (e.g., when R is Z(n) or Z(n)[i]), when R is a polynomial or a power series ring, and when R is the ring of all real-valued continuous functions on a topological space.