Semirings which have linearly ordered prime ideals

As a generalization of valuation semirings, the main purpose of this paper is to investigate those semirings that their prime ideals are totally ordered by inclusion. First, we prove that the prime ideals of a semiring $S$ are linearly ordered if and only if for each $x,y \in S$, there is a positive integer $n$ such that either $x|y^n$ or $y|x^n$. Then we introduce and characterize pseudo-valuation semidomains. It is shown that prime ideals of pseudo-valuation semidomains and also divided ones are linearly ordered.Comment: Some new references added. Some minor typos edite

On the Content of Polynomials Over Semirings and Its Applications

In this paper, we prove that Dedekind-Mertens lemma holds only for those semimodules whose subsemimodules are subtractive. We introduce Gaussian semirings and prove that bounded distributive lattices are Gaussian semirings. Then we introduce weak Gaussian semirings and prove that a semiring is weak Gaussian if and only if each prime ideal of this semiring is subtractive. We also define content semialgebras as a generalization of polynomial semirings and content algebras and show that in content extensions for semirings, minimal primes extend to minimal primes and discuss zero-divisors of a content semialgebra over a semiring who has Property (A) or whose set of zero-divisors is a finite union of prime ideals. We also discuss formal power series semirings and show that under suitable conditions, they are good examples of weak content semialgebras.Comment: Final version published at J. Algebra Appl., one reference added, three minor editorial change

Zero-divisor graphs of nilpotent-free semigroups

We find strong relationships between the zero-divisor graphs of apparently disparate kinds of nilpotent-free semigroups by introducing the notion of an \emph{Armendariz map} between such semigroups, which preserves many graph-theoretic invariants. We use it to give relationships between the zero-divisor graph of a ring, a polynomial ring, and the annihilating-ideal graph. Then we give relationships between the zero-divisor graphs of certain topological spaces (so-called pearled spaces), prime spectra, maximal spectra, tensor-product semigroups, and the semigroup of ideals under addition, obtaining surprisingly strong structure theorems relating ring-theoretic and topological properties to graph-theoretic invariants of the corresponding graphs.Comment: Expanded first paragraph in section 6. To appear in J. Algebraic Combin. 22 page