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Beck′s Coloring of a Commutative Ring

✍ Scribed by D.D. Anderson; M. Naseer

Publisher
Elsevier Science
Year
1993
Tongue
English
Weight
574 KB
Volume
159
Category
Article
ISSN
0021-8693

No coin nor oath required. For personal study only.

✦ Synopsis

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 (x y=0). Beck conjectured that (\chi(R)=c l(R)). We give a counterexample where (R) is a finite local ring with (c l(R)=5) but (\chi(R)=6). We show that if (A) is a regular Noetherian ring with maximal ideals (N_{1}, \ldots, N_{s}) such that each (A / N_{i}) is finite, then for (R=A / N_{1}^{n_{1}} \cdots N_{s}^{n_{s}}, \chi(R)=c /(R)). Finally, we give a complete listing of all finite rings (R) with (\chi(R) \leqslant 4). 1993 Academic Press, Inc.

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