mersenneforum.org Discrete Ordered Rings?
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 2013-11-25, 04:48 #1 jinydu     Dec 2003 Hopefully Near M48 33368 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 3×7×53 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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