Talk:Ordered ring

Positive elements
I don't have a good idea how to do that, but it should be made clear that in the properties $$R_+$$ is the set of positive elements not nonnegative as in "or, in some cases, nonnegative". Otherwise property 3 is not true and property 4 has a superfluous assumption about zero divisors. — Preceding unsigned comment added by Slawekk (talk • contribs) 25 December 2005

Positive cones
There is a fair amount of discussion on the ordered field entry about positive cones and the equivalence between the 'field with a compatible total order' definition and the 'field with a positive cone' definition. I'm pretty sure that the same equivalence applies here as well. Perhaps that discussion could be either moved here or factored into a separate page on positive cones?

Mdgeorge (talk) 21:01, 17 November 2012 (UTC)

Ordered rings are not necessarily commutative
I am very surprised by the first sentence of this article. I've been reading about noncommutative ordered rings for some time, for example in "A First Course in Noncommutative Rings" by Tsi-Yuen Lam. Here's what appears on page 272. Historically, the first example of a noncommutative ordered division ring was constructed by Hilbert (in 1903), in connection with his study of the foundations of geometry. Hilbert's example, based on the use of twisted Laurent series, was later generalized by Mal'cev and Neumann in 1948–1949.

So surely the first sentence of this wikipedia article is wrong! Could someone please fix this? I am not an algebraist. — Preceding unsigned comment added by Alan U. Kennington (talk • contribs) 05:09, 11 March 2014 (UTC)

Consequence of assuming a total order
"Exactly one of the following is true: a is positive, -a is positive, or a = 0. This property follows from the fact that ordered rings are abelian, linearly ordered groups with respect to addition."

In fact it seems to follow directly from the assumption of a total order (specifically the connex property).— Pingkudimmi 09:27, 28 August 2018 (UTC)