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Minimal prime ideals and cycles in annihilating-ideal graphs

Aalipour, G ; Sharif University of Technology | 2013

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  1. Type of Document: Article
  2. DOI: 10.1216/RMJ-2013-43-5-1415
  3. Publisher: 2013
  4. Abstract:
  5. Let R be a commutative ring with identity, and let 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}, and two distinct vertices I and J are adjacent if and only if IJ = 0. In this paper, we study some connections between the graph theoretic properties of this graph and some algebraic properties of a commutative ring. We prove that if AG(R) is a tree, then either AG(R) is a star graph or a path of order 4 and, in the latter case, R = F × S, where F is a field and S is a ring with a unique non-trivial ideal. Moreover, we prove that if R has at least three minimal prime ideals, then AG(R) is not a tree. It is shown that, for every reduced ring R, if R has at least three minimal prime ideals, then AG(R) contains a triangle. Also, we prove that, for every non-reduced ring R, if |Min(R)| = 2, then either AG(R) contains a cycle or AG(R) = P4. Finally, it is proved that, if |Min (R)| = 1 and AG(R) is a bipartite graph, then AG(R) is a star graph
  6. Keywords:
  7. Annihilating-ideal graph ; Bipartite graph ; Cycle ; Minimal prime ideal
  8. Source: Rocky Mountain Journal of Mathematics ; Volume 43, Issue 5 , 2013 , Pages 1415-1425 ; 00357596 (ISSN)
  9. URL: http://www.projecteuclid.org/euclid.rmjm/1382705659