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The Maximal Regular Ideal of Some Commutative Rings
All HMC Faculty Publications and Research
  • Emad Abu Osba, University of Jordan
  • Melvin Henriksen, Harvey Mudd College
  • Osama Alkam, University of Jordan
  • Frank A. Smith, Kent State University
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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 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.

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© 2006 Charles University in Prague
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Citation Information
Osba, 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.