On the primality and elasticity of algebraic valuations of cyclic free semirings

半环诱导赋值代数同构与半环同构之间的关系

On the Relationship Between Semiring-Induced Valuation Algebra Isomorphisms and Semiring Isomorphisms

本文探究半环同构与诱导赋值代数同构的双向关系，证明半环同构可推出赋值代数同构，且对特定半环，赋值代数同构也能推出底层半环同构。

Some remarks on the comparability of ideals in semirings

A semiring is uniserial if its ideals are totally ordered by inclusion. First, we show that a semiring $S$ is uniserial if and only if the matrix semiring $M_n(S)$ is uniserial. As a generalization of valuation semirings, we also investigate those semirings whose prime ideals are linearly ordered by inclusion. For example, we prove that the prime ideals of a commutative 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 prim…

Valuation Semirings

The main scope of this paper is to introduce valuation semirings in general and discrete valuation semirings in particular. In order to do that, first we define valuation maps and investigate them. Then we define valuation semirings with the help of valuation maps and prove that a multiplicatively cancellative semiring is a valuation semiring if and only if its ideals are totally ordered by inclusion. We also prove that if the unique maximal ideal of a valuation semiring is subtractive, then it is integrally closed. We end this paper by introducing discrete valuation semirings and show that a …

环的谱及其理想半环的一个泛性质

A Universal Property of the Spectrum of a Ring and the Semiring of its Ideals

该研究利用广义幂等半环赋值明确了算术赋值与交换环理想结构的关联，证明理想半环是正赋值的泛分解半环，并将环谱的泛性质与算术赋值理论联系起来，给出闭Zariski拓扑的泛分解刻画。

Valuations of Semirings

We develop notions of valuations on a semiring, with a view toward extending the classical theory of abstract nonsingular curves and discrete valuation rings to this general algebraic setting; the novelty of our approach lies in the implementation of hyperrings to yield a new definition (\emph{hyperfield valuation}). In particular, we classify valuations on the semifield $\mathbb{Q}_{max}$ (the max-plus semifield of rational numbers) and also valuations on the `function field' $\mathbb{Q}_{max}(T)$ (the semifield of rational functions over $\mathbb{Q}_{max}$) which are trivial on $\mathbb{Q}_{…

关于多项式半环代数赋值的加法结构II

On the additive structure of algebraic valuations of polynomial semirings II

研究多项式半环\(\mathbb{N}_0[\alpha]\)加法结构，从赋值理论角度出发，证明\(\mathbb{N}_0[\alpha]\)加法幺半群无加法不可约元的充要条件，描述相关代数参数\(\alpha\)，并论证特定\(\alpha\)子集在\((0,1)\)中稠密。