Some non-chromatic rings

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