Year: 2016
Type: article
Abstract: The annihilating-ideal graph of a commutative ring R, denoted by ๐ธ๐พ(R), is a graph whose vertex set consists of all non-zero annihilating ideals and two distinct vertices I and J are adjacent if and only if IJ = (0). Here we show that if R is a reduced ring and the independence number of ๐ธ๐พ(R) is finite, then the edge chromatic number of ๐ธ๐พ(R) equals its maximum degree and this number equals 2|Min(R)|โ1 also, it is proved that the independence number of ๐ธ๐พ(R) equals 2|Min(R)|โ1, where Min(R) denotes the set of minimal prime ideals of R. Then we give some criteria for a graph to be isomorphic with an annihilating-ideal graph of a ring. For example, it is shown that every bipartite annihilating-ideal graph is a complete bipartite graph with at most two horns. Among other results, it is shown that a รปnite graph ๐ธ๐พ(R) is not Eulerian, and that it is Hamiltonian if and only if R contains no Gorenstain ring as its direct summand.
Source: Canadian Mathematical Bulletin
Author: Farzad Shaveisi
Institution: Razi University
Language: English
Topic: Rings, Modules, and Algebras
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W2531842724