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 jinydu 2013-11-25 04:48

Discrete Ordered Rings?

Are there any examples other than [tex]\mathbb{Z}[/tex]?

(Alright, alright. I know of one other example: [tex]\prod_U\mathbb{Z}[/tex], i.e. ultrapowers of [tex]\mathbb{Z}[tex]. But this is not helpful for the problem I've got.)

 jinydu 2013-11-25 09:01

Never mind, found an example: The ring of polynomials (in one variable) with natural number coefficients.

 NBtarheel_33 2013-11-25 09:08

[URL="http://matwbn.icm.edu.pl/ksiazki/fm/fm85/fm85114.pdf"]This[/URL] 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 [I]R[/I] is an ordered ring with unity and if [I]a[/I] is an element of [I]R[/I], then the order in [I]R[/I] extends to the ring of polynomials [TEX]S = R[x] / <(x - a)^2>[/TEX]. [B]Moreover, if [I]R[/I] is discrete, then so is [I]S[/I]. [/B](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 [TEX]Z[x] / <(x - 8)^2>[/TEX], 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! :smile:

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