The Maximal Regular Ideal of Some Commutative Rings

Authors

Emad Abu Osba, University of JordanFollow
Melvin Henriksen, Harvey Mudd CollegeFollow
Osama Alkam, University of JordanFollow
Frank A. Smith, Kent State UniversityFollow

Document Type

Article

Department

Mathematics (HMC)

Publication Date

2006

Abstract

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 ℤ_n or 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.

Comments

Previously linked to as: http://ccdl.libraries.claremont.edu/u?/irw,435.

Article can also be found at http://dml.cz/dmlcz/119569

Rights Information

© 2006 Charles University in Prague

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Terms of Use for work posted in Scholarship@Claremont.

Recommended Citation

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.