# Questions tagged [ordered-rings]

Ordered rings are (usually commutative) rings which have an additional structure, a linear order compatible with the ring structure. This tag is for questions regarding ordered rings and their properties, as well proofs related to un-orderability of certain rings.

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### Is the class of linearly-orderable rings first order axiomatizable?

A linearly ordered ring is a commutative ring $R$ with unity equipped with a linear order $\leq$ that is compatible with addition, and such that the set of nonnegative elements are closed under ...
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### Can a unique factorisation domain have a largest prime?

Suppose $R$ is a UFD and $(R,\leq)$ is an ordered ring. Is it possible that $R$ has a largest prime element? Below is my attempt so far to answer this myself, though I'm still unsure what the ...
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1 vote
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### Visualize the Completion of (the Ordered Field) of Rational Functions

Every ordered abelian group $G$ can be completed to give a larger ordered abelian group $\bar{G}$. The original abelian group $G$ embeds into $\bar{G}$ as a dense subset, and every non-empty subset of ...
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### Multiplication Law for Order on Integers

I'm using the following definitiosn for addition $+$, multiplication $\cdot$, and the relation $\preceq$ on the set of integers: \begin{align*}\tag{I} [(a,b)]+[(c,d)]&:=[(a+c,b+d)] \\ \tag{II} [(a,...
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### Does an infinite chain contain articulation points?

I had a question which asked whether 2-regular graphs have any articulation points. We assumed finite graphs so it's just a disjoint union of cycles. However if we allow infinite graphs how do we ...
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### Is the number of orderings the same of the number of automorphisms in a ring?

Q: Given an ordered ring $A$ is the number of automorphisms of $A$ equal to the number of orderings in $A$? An ordering on a ring is totally defined by a subset of $A$ we call $A^+$ that satisfies ...
1 vote
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### Is it possible to send an element in an ordered real algebra both to a positive unit and to a negative unit?

Let ${(A, P)}$ be a preordered $\mathbb{R}$-algebra in the sense that $A$ is a $\mathbb{R}$-algebra and ${P \subseteq A}$ is a subset closed under addition, multiplication, containing the nonnegative ...
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### An example of an ordered UFD except the ring of integers?

Are there examples of unique factorization domains which are ordered rings https://en.m.wikipedia.org/wiki/Ordered_ring except the ring of integers?
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### Metric mapping to sets other than $\Bbb{R}$ [duplicate]

A metric space is a set M together with a function $d:M \times M \rightarrow \Bbb{R}$, where $d$ satisfies: $d(x,y)\ge 0$ $d(x,y)=0 \Leftrightarrow x=y$ $d(x,y)=d(y,x)$ $d(x,z) \le d(x,y)+ d(y,z)$ ...
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If $(E,<)$ is a linear order, let $s(E)$ denote the supremum of the set of ordinals which (order-)embed in $(E,<)$. $s(E)$ is also the set of ordinals which embed in $E$ with a non cofinal range....