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# Article

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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-Čech compactification; \$C(X)\$; zerosets; cozerosets; \$P\$-spaces
Summary:
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.
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