Beck′s Coloring of a Commutative Ring
Beck′s Coloring of a Commutative Ring
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DOI：
10.1006/jabr.1993.1171
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发表时间：
1993-08
影响因子：
0.9
通讯作者：
D. D. Anderson-D.;M. Naseer
中科院分区：
文献类型：
--
作者：
D. D. Anderson-D.;M. Naseer
交换环R可以看作是一个简单的图，其顶点是R中的元素，R中的两个不同的元素x和y相邻当且仅当xy = 0。Beck推测χ(R) = cl(R)。我们给出了一个反例，其中R是cl(R) = 5但χ(R) = 6的有限局部环。我们证明了如果A是一个极大理想N1的正则noether环，…，使得每个A/Ni是有限的，则对于R = A/Nn11···Nnss, χ(R) = cl(R)。最后，我们给出了χ(R)≤4的所有有限环R的完整列表。
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 N1, ..., Ns, such that each A/Ni is finite, then for R = A/Nn11 ··· Nnss, χ(R) = cl(R). Finally, we give a complete listing of all finite rings R with χ(R) ≤ 4.