Journal of the
Korean Mathematical Society JKMS

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

MSC numbers: 16Y70, 05C62