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On the complement of the intersection graph of submodules of a module

Akbari, S ; Sharif University of Technology | 2015

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  1. Type of Document: Article
  2. DOI: 10.1142/S0219498815501169
  3. Publisher: World Scientific Publishing Co. Pte Ltd , 2015
  4. Abstract:
  5. Let R be a ring with identity and M be a unitary left R-module. The complement of the intersection graph of submodules of M, denoted by Γ(M), is defined to be a graph whose vertices are in one-to-one correspondence with all nontrivial submodules of M and two distinct vertices are adjacent if and only if the corresponding submodules of M have zero intersection. In this paper, we consider the complement of the intersection graph of submodules of a module. We prove that, if Γ(M) is connected and Δ(Γ(M)) < ∞, then M is semisimple, where Δ(Γ(M)) is the maximum degree of Γ(M). We show that, if Γ(M) is a forest, then each component of Γ(M) is a star graph. Moreover, it is proved that, if Γ(M) is a tree, then Γ(M) is isomorphic to a complete graph of order at most two
  6. Keywords:
  7. Complement of the intersection graph ; Module
  8. Source: Journal of Algebra and its Applications ; Volume 14, Issue 8 , October , 2015 ; 02194988 (ISSN)
  9. URL: http://www.worldscientific.com/doi/10.1142/S0219498815501169