Let $R$ be a not necessarily commutative ring with $1$. In the present paper we first introduce a notion of quasi-orderings, which axiomatically subsumes all the orderings and valuations on $R$. We proceed by uniformly defining a coarsening relation $\leq$ on the set $\mathcal{Q}(R)$ of all quasi-orderings on $R$. One of our main results states that $(\mathcal{Q}(R), \leq^\prime)$ is a rooted tree for some slight modification $\leq^\prime$ of $\leq$, i.e. a partially ordered set admitting a maximum such that for any element there is a unique chain to that maximum. As an application of this theorem we obtain that $(\mathcal{Q}(R), \leq^\prime)$ is a spectral set, i.e. order-isomorphic to the spectrum of some commutative ring with $1$. We conclude this paper by studying $\mathcal{Q}(R)$ as a topological space.

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关于环的订购和估价的树形结构

令$ R $不一定是与$ 1 $的可交换环。在本文中，我们首先介绍一个准排序的概念，它在公理上将$ R $的所有排序和估值都包含在内。我们通过在$ R $上所有拟排序的集合$ \ mathcal {Q}（R）$上统一定义一个粗化关系$ \ leq $。我们的主要结果之一表明，$（\ mathcal {Q}（R），\ leq ^ \ prime）$是对$ \ leq $进行一些轻微修改$ \ leq ^ \ prime $的根树，即部分排序设置允许最大值，以便对于任何元素，都有一个唯一的链达到该最大值。作为该定理的一个应用，我们得到$（\ mathcal {Q}（R），\ leq ^ \ prime）$是一个谱集，即与具有$ 1 $的某些可交换环的谱是同构的。我们通过研究$ \ mathcal {Q}（R）$作为拓扑空间来结束本文。