Some non-chromatic rings

Bhatwadekar, S. M. ; Dumaldar, Mahesh N. ; Sharma, Pramod K. (1998) Some non-chromatic rings Communications in Algebra, 26 (2). pp. 477-505. ISSN 0092-7872

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Let R be a commutative ring with identity. Consider R as a simple graph with vertices elements of R where any two distinct elements x, y ε R are adjacent if and only if xy = 0. Beck [2] conjectured that χ(R) =c1(R), for all colorings R. Anderson and Naseer [1] provided a counter example to the conjecture i.e., gave an example of a non-chromatic ring. This paper generalizes the example and gives a large class of non-chromatic rings [Theorem 1.1]. A detailed analysis of the graph of these rings is also provided.

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