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.
The Maximal Regular Ideal of Some Commutative RingsAll HMC Faculty Publications and Research
Rights Information© 2006 Charles University in Prague
Citation InformationOsba, Emad Abu, Melvin Henriksen, Osama Alkam, and F. A. Smith. "The maximal regular ideal of some commutative rings." Commentationes Mathematicae Universitatis Carolinae 47.1 (2006): 1-10.