# 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 every ordered abelian group the additive group of an ordered ring?

Let $\Lambda$ be an ordered abelian group, (there is a total order on $\Lambda$ which is compatible with addition). Is there a multiplication map on $\Lambda$ that turns it into an ordered ring? I ...
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### Are $\mathbb{R}$ and $\mathbb{Q}$ the only subfields of $\mathbb{C}$ with natural structure as ordered fields?

We know that $\mathbb{R}$ and $\mathbb{Q}$ have a unique structure as ordered fields with the usual order, and that $\mathbb{C}$ cannot be realised as an ordered field. Various non-trivial subfields ...
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### $a^+$ Induced Order

I am studying the different orders that can be induced from the usual polynomial order in $\mathbb R[x]$: i.e: $$p(x) >_{+\infty} 0 \iff a_n > 0$$ Where $a_n$ is the leading coefficient. One ...
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### Greateat common divisor in Z[i]

How do i use the euclidean algorithm to compute the greatest common divisor of two elements in Z[i]? 