• Corpus ID: 119324334

# The coloring of the regular graph of ideals

@article{Shaveisi2015TheCO,
title={The coloring of the regular graph of ideals},
journal={arXiv: Combinatorics},
year={2015}
}
• F. Shaveisi
• Published 2 January 2015
• Mathematics
• arXiv: Combinatorics
The regular graph of ideals of the commutative ring $R$, denoted by $\Gamma_{reg}(R)$, is a graph whose vertex set is the set of all non-trivial ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if either $I$ contains a $J$-regular element or $J$ contains an $I$-regular element. In this paper, it is shown that for every Artinian ring $R$, the edge chromatic number of $\Gamma_{reg}(R)$ equals its maximum degree. Then a formula for the clique number of $\Gamma_{reg}(R… ## References SHOWING 1-10 OF 16 REFERENCES The regular digraph of ideals of a commutative ring • Mathematics • 2012 AbstractLet R be a commutative ring and Max (R) be the set of maximal ideals of R. The regular digraph of ideals of R, denoted by$\overrightarrow{\Gamma_{\mathrm{reg}}}(R)$, is a digraph whose The Annihilating-Ideal Graph of Commutative Rings I • Mathematics • 2008 Let$R$be a commutative ring with${\Bbb{A}}(R)$its set of ideals with nonzero annihilator. In this paper and its sequel, we introduce and investigate the {\it annihilating-ideal graph} of$R\$,
Minimal prime ideals and cycles in annihilating-ideal graphs
• Mathematics
• 2013
Let R be a commutative ring with identity and A(R) be the set of ideals with non-zero annihilator. The annihilating-ideal graph of R is defined as the graph AG(R) with the vertex set A(R)∗ = A(R)\{0}
THE TOTAL TORSION ELEMENT GRAPH WITHOUT THE ZERO ELEMENT OF MODULES OVER COMMUTATIVE RINGS
Abstract. Let M be a module over a commutative ring R, and let T(M)be its set of torsion elements. The total torsion element graph of Mover R is the graph T(Γ(M)) with vertices all elements of M, and
An Ideal-Based Zero-Divisor Graph of a Commutative Ring
Abstract For a commutative ring R with identity, the zero-divisor graph of R, denoted Γ(R), is the graph whose vertices are the non-zero zero-divisors of R with two distinct vertices joined by an
The minimal prime spectrum of a reduced ring
Throughout this discussion R will be a commutative ring with 1. We say R is a reduced ring if it has no nilpotent elements other than 0. Of course, this is equivalent, to saying that the intersection
A GENERALIZATION OF THE ZERO-DIVISOR GRAPH FOR MODULES
• Mathematics
• 2014
Abstract. Let R be a commutative ring with identity and M an R-module. In this paper, we associate a graph to M, say Γ(M), such thatwhen M = R, Γ(M) is exactly the classic zero-divisor graph. Many
TOTAL GRAPH OF A COMMUTATIVE SEMIRING WITH RESPECT TO IDENTITY-SUMMAND ELEMENTS
• Mathematics
• 2014
Abstract. Let R be an I-semiring and S(R) be the set of all identity-summand elements of R. In this paper we introduce the total graphof R with respect to identity-summand elements, denoted by
Cohen-Macaulay rings
• Mathematics
• 1993
In this chapter we introduce the class of Cohen–Macaulay rings and two subclasses, the regular rings and the complete intersections. The definition of Cohen–Macaulay ring is sufficiently general to