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On the coloring of the annihilating-ideal graph of a commutative ring
Aalipour, G ; Sharif University of Technology | 2012
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- Type of Document: Article
- DOI: 10.1016/j.disc.2011.10.020
- Publisher: Elsevier , 2012
- Abstract:
- Suppose that R is a commutative ring with identity. Let A(R) be the set of all ideals of R with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph AG(R) with the vertex set A( R)*=A(R)(0) and two distinct vertices I and J are adjacent if and only if IJ=(0). In Behboodi and Rakeei (2011) [8], it was conjectured that for a reduced ring R with more than two minimal prime ideals, girth(AG(R))=3. Here, we prove that for every (not necessarily reduced) ring R, ω(AG(R))<|Min(R)|, which shows that the conjecture is true. Also in this paper, we present some results on the clique number and the chromatic number of the annihilating-ideal graph of a commutative ring. Among other results, it is shown that if the chromatic number of the zero-divisor graph is finite, then the chromatic number of the annihilating-ideal graph is finite too. We investigate commutative rings whose annihilating-ideal graphs are bipartite. It is proved that AG(R) is bipartite if and only if AG(R) is triangle-free
- Keywords:
- Annihilating-ideal graph ; Chromatic number ; Clique number ; Minimal prime ideal
- Source: Discrete Mathematics ; Volume 312, Issue 17 , 2012 , Pages 2620-2626 ; 0012365X (ISSN)
- URL: http://www.sciencedirect.com/science/article/pii/S0012365X11004778