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divisor theory

Defines: 
divisor, prime divisor, principal divisor, unit divisor
Keywords: 
prime factorization
Major Section: 
Reference
Type of Math Object: 
Definition

Mathematics Subject Classification

11A51 no label found13A05 no label found

Comments

Divisor theory looks very interesting. Do you have any examples of domains having a divisor theory which aren't PIDs or Dedekind domains?
Z[X] is a UFD, but not a PID, so prime factorization in Z[X] is not a valid divisor theory. Does it have one?

Dear gel,
I cannot answer to your question -- about forty years have run after I studied the divisors. I took the material of my divisor entries from my old study work, from Postnikov's little book and from Borewicz--Shafarevic. Perhaps in some exercises of the latter, you can find some examples you want.
(Cf. http://planetmath.org/encyclopedia/ImageIdealOfDivisor.html)
Regards,
Jussi

in fact, why isn't factorization in Z[X] a divisor theory? It seems to satisfy the requirements 1-3 listed in this entry, but all divisors are principal, which would contradict Theorem 2 that such domains are PIDs. Did I miss something here?

Theorem 2 says that such domains are UFD's, not PID's.

Sorry, I misread it. So factorization in Z[X] is a valid divisor theory.

ok, I don't have those books, but now my confusion over Z[X] is sorted out, I think R[X] for R the ring of integers in a number field will have a division algebra but isnt a UFD or Dedekind domain in all cases.

You certainly find the books at library. I recommend Borewicz--Shafarevic!

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