Beck′s Coloring of a Commutative Ring

Some Properties of the Complement of the Zero-Divisor Graph of a Commutative Ring

“…In such a case, it was shown in the proof of 18, Proposition 1.7 i that there exist a ∈ P 1 \ P 2 and b ∈ P 2 \ P 1 such that ab / 0. It follows that P 1 / 0 : R a , P 1 / 0 : R b , and P 2 / 0 : R a , P 2 Suppose that R has exactly two maximal N-primes of 0 and both are B-primes of 0 in R. Let {P 1 , P 2 } be the set of all maximal N-primes of 0 in R. Then it follows from Lemmas 2.1, 2.2, and 2.4 that the center of Γ R c {x ∈ Z R * | P 1 / 0 : R x and P 2 / 0 : R x }.…”

The ideal-based zero-divisor graph of commutative chained rings

“…In his work, all elements of R were vertices of the graph, and distinct vertices x and y were adjacent if and only if xy = 0. This investigation of colorings of a commutative ring was then continued by D. D. Anderson and M. Naseer in [1]. Let Z(R) be the set of zero-divisors of R. In [5], D. F. Anderson and P. S. Livingston associated a (simple) graph Γ(R) to R, with vertices Z(R) * = Z(R)\{0}, the set of nonzero zero-divisors of R, and distinct vertices x and y are adjacent if and only if xy = 0.…”

“…In the literature, there are many papers on assigning a graph to a ring (see, for example, [1] - [7], [9], and [11]). Among the most interesting graphs are zero-divisor graphs, because they involve both ring theory and graph theory.…”

Reducing Redundancy of Codons through Total Graph

“…In his work all elements of the ring were vertices of the graph. The investigation of colorings of a commutative ring was then continued by Anderson and Naseer (1993). Anderson and Livingston (1999) associate a graph, Γ(R), to R with vertices Z (R)\{0}, where Z (R) is the set of zerodivisors of R and for distinct x, y ∈ Z (R)\{0}, the vertices x and y are adjacent if and only if xy = 0.…”