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THE IDENTITY-SUMMAND GRAPH OF COMMUTATIVE SEMIRINGS

  • Atani, Shahabaddin Ebrahimi (Faculty of Mathematical Sciences University of Guilan) ;
  • Hesari, Saboura Dolati Pish (Faculty of Mathematical Sciences University of Guilan) ;
  • Khoramdel, Mehdi (Faculty of Mathematical Sciences University of Guilan)
  • Received : 2013.05.07
  • Published : 2014.01.01

Abstract

An element r of a commutative semiring R with identity is said to be identity-summand if there exists $1{\neq}a{\in}R$ such that r+a = 1. In this paper, we introduce and investigate the identity-summand graph of R, denoted by ${\Gamma}(R)$. It is the (undirected) graph whose vertices are the non-identity identity-summands of R with two distinct vertices joint by an edge when the sum of the vertices is 1. The basic properties and possible structures of the graph ${\Gamma}(R)$ are studied.

Keywords

I-semiring;co-ideal;Q-strong co-ideal;co-semidomain;identity-summand graph;identity-summand element

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Cited by

  1. Semisimple semirings with respect to co-ideals theory 2017, https://doi.org/10.1142/S1793557118500699
  2. TOTAL IDENTITY-SUMMAND GRAPH OF A COMMUTATIVE SEMIRING WITH RESPECT TO A CO-IDEAL vol.52, pp.1, 2015, https://doi.org/10.4134/JKMS.2015.52.1.159
  3. TOTAL GRAPH OF A COMMUTATIVE SEMIRING WITH RESPECT TO IDENTITY-SUMMAND ELEMENTS vol.51, pp.3, 2014, https://doi.org/10.4134/JKMS.2014.51.3.593