# TOTAL IDENTITY-SUMMAND GRAPH OF A COMMUTATIVE SEMIRING WITH RESPECT TO A CO-IDEAL

• 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 : 2014.05.09
• Published : 2015.01.01
• 64 9

#### Abstract

Let R be a semiring, I a strong co-ideal of R and S(I) the set of all elements of R which are not prime to I. In this paper we investigate some interesting properties of S(I) and introduce the total identity-summand graph of a semiring R with respect to a co-ideal I. It is the graph with all elements of R as vertices and for distinct x, $y{\in}R$, the vertices x and y are adjacent if and only if $xy{\in}S(I)$.

#### Keywords

strong co-ideal;total identity-summand graph;identity-summand graph based a co-ideal;total identity-summand graph based a co-ideal

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