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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
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.
References:
[AHA04] Abu Osba E., Henriksen M., Alkam O.: Combining local and von Neumann regular rings. Comm. Algebra 32 (2004), 2639-2653. MR 2099923
[AM69] Atiyah M., Macdonald J.: Introduction to Commutative Algebra. Addison-Wesley, Reading, Mass., 1969. MR 0242802 | Zbl 0238.13001
[B81] Brewer J.: Power Series Over Commutative Rings. Marcel Dekker, New York, 1981. MR 0612477 | Zbl 0476.13015
[BM50] Brown B., McCoy N.: The maximal regular ideal of a ring. Proc. Amer. Math. Soc. 1 (1950), 165-171. MR 0034757 | Zbl 0036.29702
[C84] Contessa M.: On certain classes of PM rings. Comm. Algebra 12 (1984), 1447-1469. MR 0744456 | Zbl 0545.13001
[DO71] DeMarco G., Orsatti A.: Commutative rings in which every maximal ideal is contained in a unique maximal ideal. Proc. Amer. Math. Soc. 30 (1971), 459-466. MR 0282962
[GJ76] Gillman L., Jerison M.: Rings of Continuous Functions. Springer, New York, 1976. MR 0407579 | Zbl 0327.46040
[H77] Henriksen M.: Some sufficient conditions for the Jacobson radical of a commutative ring with identity to contain a prime ideal. Portugaliae Math. 36 (1977), 257-269. MR 0597848 | Zbl 0448.13002
[L58] Leveque W.: Topics in Number Theory. Addison-Wesley, Reading, Mass., 1958. Zbl 1009.11001
[M74] McDonald B.R.: Finite Rings with Identity. Marcel Dekker, New York, 1974. MR 0354768 | Zbl 0294.16012
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