Let $R$ be a commutative semiring and $I$ be a $k$-ideal of $R$. In this paper, we introduce the $k$-ideal-based graph of $R$, denoted by $\Gamma_{I^{*}}(R)$. The basic properties and possible structures of the graph are studied.

The notion and some properties of (strongly) B-rings, in a natural way, are extended to (strongly) B-and (strongly) BJ-semirings which is somewhat similar to the notion of rings having stable range 2. Results are given showing the connection between several types of semirings whose finite sequences satisfy some stability condition, some involving the Jacobson k-radical of the semiring R. Besides some examples and other results, it is shown that R[x], the semiring of polynomials over a semiring R, is not a B-semiring (consequently, not a strongly B-semiring) when R is a zerosumfree semiring. We also study some algebraic properties of the S-relative B-and BJ-semirings with respect to a nonempty subset S of R.

The concept of Γ-semihyperrings is a generalization of a semiring, a generalization of a Γ-semiring, and a generalization of a semihyperering. In this paper, we define the notions of complex product, extension property and flat Γ-semihyperrings and some of their properties are obtained. In addition, we prove that every flat Γ-semihyperring is absolutely extendable. Finally, we give some characterization of stable elements.

In a ternary semihyperring, addition is a hyperoperation and multiplication is a ternary operation. Indeed, the notion of ternary semihyperrings is a generalization of semirings. Our main purpose of this paper is to introduce the notions of fuzzy hyperideal and fuzzy bi-hyperideal in ternary semihyperrings. We give some characterizations of fuzzy hyperideals and investigate several kinds of them.