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 M (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 M(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 ℤn or ℤn[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