Geodesic
The maximal regular ideal of some commutative rings
Commentationes Mathematicae Universitatis Carolinae, Tome 47 (2006) no. 1, pp. 1-10.

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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 $\frak M (R)$ 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.
Classification : 10A10, 13A15, 13Fxx, 16E50, 54G10
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 
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Osba, Emad Abu; Henriksen, Melvin; Alkam, Osama; Smith, F. A. The maximal regular ideal of some commutative rings. Commentationes Mathematicae Universitatis Carolinae, Tome 47 (2006) no. 1, pp. 1-10. http://geodesic.mathdoc.fr/item/CMUC_2006__47_1_a0/