mersenneforum.org Discrete Ordered Rings?
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 2013-11-25, 04:48 #1 jinydu     Dec 2003 Hopefully Near M48 2·3·293 Posts Discrete Ordered Rings? Are there any examples other than $\mathbb{Z}$? (Alright, alright. I know of one other example: $\prod_U\mathbb{Z}$, 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 2013-11-25 at 04:52
 2013-11-25, 09:01 #2 jinydu     Dec 2003 Hopefully Near M48 2·3·293 Posts Never mind, found an example: The ring of polynomials (in one variable) with natural number coefficients.
 2013-11-25, 09:08 #3 NBtarheel_33     "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 $S = R[x] / <(x - a)^2>$. 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 $Z[x] / <(x - 8)^2>$, 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 2013-11-25 at 09:11

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