Abstract: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 $\frak M (R)$ 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 $\frak M (R)$ 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 $\Bbb Z_{n}$ or $\Bbb Z_{n}[i]$), 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.
Keywords: commutative rings, von Neumann regular rings, von Neumann local rings, Gelfand rings, polynomial rings, power series rings, rings of Gaussian integers (mod $n$), prime and maximal ideals, maximal regular ideals, pure ideals, quadratic residues, Stone-\v Cech compactification, $C(X)$, zerosets, cozerosets, $P$-spaces
AMS Subject Classification: 13A, 13FXX, 54G10, 10A10