Journal of the
Korean Mathematical Society JKMS

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-sum\-mand graph based a co-ideal, total identity-summand graph based a co-ideal

MSC numbers: Primary 16Y60, 05C62