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Some results on the intersection graph of submodules of a module

Akbari, S ; Sharif University of Technology | 2017

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  1. Type of Document: Article
  2. DOI: 10.1515/ms-2016-0267
  3. Publisher: De Gruyter Open Ltd , 2017
  4. Abstract:
  5. Let R be a ring with identity and M be a unitary left R-module. The intersection graph of submodules of M, denoted by G(M), is defined to be a graph whose vertices are in one to one correspondence with all non-trivial submodules of M and two distinct vertices are adjacent if and only if the corresponding submodules of M have non-zero intersection. In this paper, we consider the intersection graph of submodules of a module. We determine the structure of modules whose clique numbers are finite. We show that if 1 < ω(G(M)) < ∞, then M is a direct sum of a finite module and a cyclic module, where ω(G(M)) denotes the clique number of G(M). We prove that if ω(G(M)) is not finite, then M contains an infinite clique. Among other results, it is shown that a Noetherian R-module whose intersection of all non-trivial submodules is non-zero, is Artinian. © 2017 Mathematical Institute Slovak Academy of Sciences
  6. Keywords:
  7. Clique number ; Intersection graph ; Module
  8. Source: Mathematica Slovaca ; Volume 67, Issue 2 , 2017 , Pages 297-304 ; 01399918 (ISSN)
  9. URL: https://www.degruyter.com/view/j/ms.2017.67.issue-2/ms-2016-0267/ms-2016-0267.xml