Journal article
Beck′s Coloring of a Commutative Ring
Journal of algebra, Vol.159(2), pp.500-514
1993
DOI: 10.1006/jabr.1993.1171
Abstract
A commutative ring R can be considered as a simple graph whose vertices are the elements of R and two different elements x and y of R are adjacent if and only if xy = 0. Beck conjectured that χ( R) = cl( R). We give a counterexample where R is a finite local ring with cl( R) = 5 but χ( R) = 6. We show that if A is a regular Noetherian ring with maximal ideals N 1, ..., N s , such that each A/ N i is finite, then for R = A/ N n 1 1 ··· N n s s , χ( R) = cl( R). Finally, we give a complete listing of all finite rings R with χ( R) ≤ 4.
Details
- Title: Subtitle
- Beck′s Coloring of a Commutative Ring
- Creators
- D.D AndersonM Naseer
- Resource Type
- Journal article
- Publication Details
- Journal of algebra, Vol.159(2), pp.500-514
- DOI
- 10.1006/jabr.1993.1171
- ISSN
- 0021-8693
- eISSN
- 1090-266X
- Publisher
- Elsevier Inc
- Language
- English
- Date published
- 1993
- Academic Unit
- Mathematics
- Record Identifier
- 9983985838902771
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