20131125, 04:48  #1 
Dec 2003
Hopefully Near M48
2×3×293 Posts 
Discrete Ordered Rings?
Are there any examples other than ?
(Alright, alright. I know of one other example: , i.e. ultrapowers of [tex]\mathbb{Z}[tex]. But this is not helpful for the problem I've got.) Last fiddled with by jinydu on 20131125 at 04:52 
20131125, 09:01  #2 
Dec 2003
Hopefully Near M48
11011011110_{2} Posts 
Never mind, found an example: The ring of polynomials (in one variable) with natural number coefficients.

20131125, 09:08  #3 
"Nathan"
Jul 2008
Maryland, USA
5×223 Posts 
This paper, entitled (appropriately enough) "Discrete Ordered Rings", might be of some help.
In particular, look at Theorem 11.1 on page 135. It states that if R is an ordered ring with unity and if a is an element of R, then the order in R extends to the ring of polynomials . Moreover, if R is discrete, then so is S. (The proof follows in the paper, and explains how the ordering works.) So, given that the integers give you a discrete ordered ring, it seems as though you could just pick your favorite integer (I like 8) and then form the polynomial ring , and that would then be yet another example of a discrete ordered ring by the above theorem. Hopefully I have understood this correctly, and this helps you out! Last fiddled with by NBtarheel_33 on 20131125 at 09:11 
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