Re: Infinite UFD, finite many units => infinitely many pairwise nonassociated primes

On Oct 5, 6:55 pm, Robert Israel
<isr...@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:
"Achava Nakhash, the Loving Snake" <ach...@xxxxxxxxxxx> writes:





On Oct 4, 11:22=A0pm, Kenneth Bull <kenneth.b...@xxxxxxxxx> wrote:
I am trying to prove that an infinite Unique Factorization Domain with
finitely many units has infinitely many pairwise nonassociated
primes. =A0Can someone give me a hint to get started?

My first thought would be to try to mimic Euclid's argument and see
where it led.  So if there are only finitely many pairwise, non-
associated primes, let them be called p_1, p_2, ..., p_n.  Now look at
a =3D 1 + their product.  It is obviously not divisible by any of these
primes nor by any of their associates.  Your finiteness conditions
certainly tell as that there are other elements in the ring than a or
than all of these associates.  Since a must have a unique
factorization into primes if it is not a unit, then it must have a
prime factor not associated with p_1, ..., p_n.  Can you think of a
construction for a that ensures that it is not a unit?

Just a thought,
Achava

Perhaps it is a unit, but only finitely many of the elements
1 + (p_1 ... p_n)^k are.
--
Robert Israel              isr...@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics        http://www.math.ubc.ca/~israel
University of British Columbia            Vancouver, BC, Canada- Hide quoted text -

- Show quoted text -

Hey, I was trying to give a hint, not solve the problem for him.

Regards,
Achava
.

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