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On the Unit Graph of a Non-commutative Ring

Akbari, S ; Sharif University of Technology | 2015

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  1. Type of Document: Article
  2. DOI: 10.1142/S100538671500070X
  3. Publisher: World Scientific Publishing Co. Pte Ltd , 2015
  4. Abstract:
  5. Let R be a ring with non-zero identity. The unit graph G(R) of R is a graph with elements of R as its vertices and two distinct vertices a and b are adjacent if and only if a + b is a unit element of R. It was proved that if R is a commutative ring and is a maximal ideal of R such that |R/| = 2, then G(R) is a complete bipartite graph if and only if (R, ) is a local ring. In this paper we generalize this result by showing that if R is a ring (not necessarily commutative), then G(R) is a complete r-partite graph if and only if (R, ) is a local ring and r = |R/| = 2n for some n ∞ N or R is a finite field. Among other results we show that if R is a left Artinian ring, 2 U(R) and the clique number of G(R) is finite, then R is a finite ring
  6. Keywords:
  7. Clique number ; Complete r-partite graph ; Unit graph
  8. Source: Algebra Colloquium ; Volume 22 , December , 2015 , Pages 817-822 ; 10053867 (ISSN)
  9. URL: http://www.worldscientific.com/doi/10.1142/S100538671500070X