# 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.

21 questions
Filter by
Sorted by
Tagged with
1 vote
46 views

### 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 ...
32 views

### Dedekind completion of the integers

The Dedekind completion of any Archimedean ordered field is the real numbers. The same is true of the Dedekind completion of any Archimedean ordered integral domain whose strict order is dense. What ...
21 views

65 views

### 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,...
1 vote
34 views

68 views

32 views

### 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 ...
338 views

### 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
55 views

### 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 ...
55 views

### 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?
487 views

### 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: $d(x,y)\ge 0$ $d(x,y)=0 \Leftrightarrow x=y$ $d(x,y)=d(y,x)$ $d(x,z) \le d(x,y)+ d(y,z)$ ...
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....