On the tree structure of orderings and valuations on rings

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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Received 14 April 2020

Received revised 24 July 2020

Accepted 4 August 2020

Published 4 May 2021