Abstract
Let R be a Noetherian ring. We denote by G(R) the simple graph whose vertices are elements of R and in which two distinct vertices x and y are joined by an edge if x − y is a zero-divisor of R. Let \({\overline{G}(R)}\) be the complementary graph to G(R) and \({\overline{\chi}(R)}\) be the chromatic number of the graph \({\overline{G}(R)}\). In this paper, we determine the chromatic number \({\overline{\chi}(R)}\). Let P1, . . . , Pt be all maximal prime divisors of (0). If \({\overline{\chi}(R)}\) is finite, then $$\overline{\chi}(R) = {\rm min} \{|R/P_{i}|; i = 1, 2, \ldots, t\}$$ where \({|R/P_{i}|}\) denotes the number of the set R/Pi.
Highlights
Let C be a non-empty subset of V (G)
Let R be a commutative ring with the identity element
We consider the simple graph G(R) whose vertices are elements of R and in which two distinct vertices x and y are joined by an edge if x − y is in Z (R)
Summary
Let C be a non-empty subset of V (G). We call C a clique of G if every pair of two distinct elements of C is joined by an edge. For a set S, |S| denotes the number of element of S. Let R be a commutative ring with the identity element. We consider the simple graph G(R) whose vertices are elements of R and in which two distinct vertices x and y are joined by an edge if x − y is in Z (R). AssR(N ) denotes the set of all associated prime ideals of N .
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