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From: juandiego on 2 Sep 2009 15:05
On 2 Sep, 17:55, Bill Dubuque <w...(a)nestle.csail.mit.edu> wrote:
> juandiego <sttscitr...(a)tesco.net> wrote:
> > On 2 Sep, 15:42, Bill Dubuque <w...(a)nestle.csail.mit.edu> wrote:
> >> juandiego <sttscitr...(a)tesco.net> wrote:
> >>>On 31 Aug, 15:13, Bill Dubuque <w...(a)nestle.csail.mit.edu> wrote:
> >>>>juandiego <sttscitr...(a)tesco.net> wrote:
>
> >>>>> If you assume that there is a last prime, i.e the primes
> >>>>> are finite in number, then w+1 is not divisble by ANY prime
> >>>>> (you have exhausted all the possible prime divisors of w+1)
> >>>>> Therefore, w+1 has only the trivial divisors 1, w+1
> >>>>> This means that w+1 is itself prime - a contradiction.
> >>>>> On the other hand, w+1 is >1 and has no prime divisors
> >>>>> This implies w+1 is a unit, but there are no units >1
> >>>>> - a contradiction. [...]
> >>>>> If there were units greater than 1, w+1 could have
> >>>>> no prime divisors and the proof would not go through.
>
> >>>> Not true.
> >>>> Thus, from this more general perspective, excluding the
> >>>> possibility that w+1 is a unit is crucial.
>
> >>> You seem to be contradicting yourself.
>
> >> No. First let's recall what I actually wrote, not the
> >> greatly condensed 3-line version that you quoted above.
>
> >>>> Not true. I had the pleasure of discovering as a teenager
> >>>> that Euclid's proof generalizes to any infinite ring with
> >>>> fewer units than elements, i.e. "fewunit" rings - see [1].
> >>>> The key idea is that Euclid's construction of a new prime
> >>>> generalizes from elements to ideals, i.e. given max ideals
> >>>> P1,...,Pk then a simple pigeonhole argument employing CRT
> >>>> implies that 1 + P1...Pk contains a nonunit, which lies in
> >>>> some maximal ideal P which, by construction, is comaximal
> >>>> (so distinct) from the prior max ideals Pi.
>
> >>>> Thus, from this more general perspective, excluding the
> >>>> possibility that w+1 is a unit is crucial. Many proofs
> >>>> of Euclid's theorem fail to explicitly exclude this case,
> >>>> so they are (technically) incomplete.
>
> >> The gist of my remark above was to dispute your claim that
> >> the proof "would not go through if there were units > 1".
> >> As I've pointed out, Euclid's proof can be generalized to
> >> infinite fewunits rings (rings with fewer units than elts).
>
> > The basis of Euclid's  original proof sems to be that
> > given any finite set of primes, the product of these primes +1
> > gives a number which is divisible by a prime not in the original set.
>
> > If r = sqrt(2), then Z(r) has UFT
>
> > r is a prime, NORM =2
> > -29 -21r is a prime or associated prime, norm =  -41
>
> > w+1 = r(-29 -21r )+1  = -41 -29r = unit
>
> > So given r, -29-21r, a finite set of primes from Euclid's
> > approach I cannot always conculde that there is an "extra" prime.
> > So in general, Euclids approach does not work.
>
> > How does what you say about fewunit rings apply to
> > "infinite" unit rings? If it doesn't, why is my claim wrong ?
>
> That ring has as many units as elts so its not a fewunit ring.
> The fewunits proof works for infinite rings with fewer units
> than elts, i.e. smaller cardinality.
>
Then why do you say "... In general 1 + pqr... can fail to
produce a new prime (or irreducible) because it may be a unit...".

See your full quote from [2] below.

"Thus generalizing Euclid's theorem to other rings will require
some hypotheses other than Euclidean, PID, UFD, etc, that are
preserved by localizations. In general 1 + pqr... can fail to
produce a new prime (or irreducible) because it may be a unit.
But we can also try 1 + d pqr... for any d in D. If all those
are units then D has a lot of units. Therefore we can eliminate
such failure by hypothesizing D has relatively few units, ""

We were only discussing that method

>
> >> Such rings may have units besides +-1, so the w+1 candidates
> >> may indeed be units. But since there are fewer units than elts
> >> a pigeonhole argument shows that at least one candidate isn't
> >> a unit, so the proof does indeed succeed. Thus even for the
> >> ring of integers, the proof can succeed without employing
> >> the fact that there are no units>1. Instead one may employ
> >> the scarcity of units in order to avoid them as need be.
>
> >>> A unit has no prime divisors. If w+1 could be either a prime,
> >>> composite or unit, then its non-divisibility by all primes or
> >>> any prime leads to no contradiction or no "extra" prime
> >>> (w+1 is a unit) and the proof would not go through.
>
> >> But it does go through for infinite fewunit rings like Z.
> >> If the proof in [1] is beyond your knowledge of algebra
> >> (e.g. it uses simple ideal theory) you can find a much
> >> simpler element-wise version in [2]. That 4-line proof
> >> should be easily comprehensible to a high-school student.
> >> Indeed, here is a simple specialization of said proof:
>
> >> THEOREM  If Z is infinite UFD with finitely many units
> >> then Z has infinitely many primes.
>
> >> PROOF  Via  contradiction.  Let  w = product of all primes of Z
> >> Since  Z  is infinite so is  1 + w Z  [ 1+wn = 1+wn'  <=>  n = n']
> >> so it must contain a nonzero nonunit  [ since only finite #units ]
> >> which has a prime factor p | 1 + w n. But  p|w => p|1  =><=  QED
>
> > Yes, very good. How do you generalize Euclid's approach to rings
> > with infinitely many units ?
>
> See the penultimate paragraph in [2]. Please don't delete my links
> when replying. That leaves the reader with dangling references.
>
> --Bill Dubuque
>
> [1]http://www.mathlinks.ro/viewtopic.php?p=1209616http://google.com/groups?selm=y8zk5f3rn4e.fsf%40nestle.csail.mit.edu
>
> [2] sci.math, 12 Apr 2005, Euclid in rings: infinitely many primeshttp://google.com/group/sci.math/msg/c502d844bdaeee1ehttp://google.com/groups?selm=y8zbr8kv5vl.fsf_-_%40nestle.csail.mit.edu
>
> Beware: cut'n'paste error in above: instead of nilradical, please
> read pseudo-radical = intersection of all nonzero prime ideals.- Hide quoted text -
>
> - Show quoted text -- Hide quoted text -
>
> - Show quoted text -

From: juandiego on 4 Sep 2009 07:56
On 3 Sep, 02:38, Bill Dubuque <w...(a)nestle.csail.mit.edu> wrote:
> juandiego <sttscitr...(a)tesco.net> wrote:
> >Bill Dubuque <w...(a)nestle.csail.mit.edu> wrote:
> >>juandiego <sttscitr...(a)tesco.net> wrote:
> >>> On 2 Sep, 15:42, Bill Dubuque <w...(a)nestle.csail.mit.edu> wrote:
> >>>> juandiego <sttscitr...(a)tesco.net> wrote:
> >>>>>On 31 Aug, 15:13, Bill Dubuque <w...(a)nestle.csail.mit.edu> wrote:
> >>>>>>juandiego <sttscitr...(a)tesco.net> wrote:
>
> >>>>>>> If you assume that there is a last prime, i.e the primes
> >>>>>>> are finite in number, then w+1 is not divisble by ANY prime
> >>>>>>> (you have exhausted all the possible prime divisors of w+1)
> >>>>>>> Therefore, w+1 has only the trivial divisors 1, w+1
> >>>>>>> This means that w+1 is itself prime - a contradiction.
> >>>>>>> On the other hand, w+1 is >1 and has no prime divisors
> >>>>>>> This implies w+1 is a unit, but there are no units >1
> >>>>>>> - a contradiction. [...]
> >>>>>>> If there were units greater than 1, w+1 could have
> >>>>>>> no prime divisors and the proof would not go through.
>
> >>>>>> Not true.
> >>>>>> Thus, from this more general perspective, excluding the
> >>>>>> possibility that w+1 is a unit is crucial.
>
> >>>>> You seem to be contradicting yourself.
>
> >>>> No. First let's recall what I actually wrote, not the
> >>>> greatly condensed 3-line version that you quoted above.
>
> >>>>   Not true. I had the pleasure of discovering as a teenager
> >>>>   that Euclid's proof generalizes to any infinite ring with
> >>>>   fewer units than elements, i.e. "fewunit" rings - see [1].
> >>>>   The key idea is that Euclid's construction of a new prime
> >>>>   generalizes from elements to ideals, i.e. given max ideals
> >>>>   P1,...,Pk then a simple pigeonhole argument employing CRT
> >>>>   implies that 1 + P1...Pk contains a nonunit, which lies in
> >>>>   some maximal ideal P which, by construction, is comaximal
> >>>>   (so distinct) from the prior max ideals Pi.
>
> >>>>   Thus, from this more general perspective, excluding the
> >>>>   possibility that w+1 is a unit is crucial. Many proofs
> >>>>   of Euclid's theorem fail to explicitly exclude this case,
> >>>>   so they are (technically) incomplete.
>
> >>>> The gist of my remark above was to dispute your claim that
> >>>> the proof "would not go through if there were units > 1".
> >>>> As I've pointed out, Euclid's proof can be generalized to
> >>>> infinite fewunits rings (rings with fewer units than elts).
>
> >>> The basis of Euclid's  original proof sems to be that given
> >>> any finite set of primes, the product of these primes +1
> >>> gives a number divisible by a prime not in the original set.
>
> >>> If r = sqrt(2), then Z(r) has UFT
>
> >>> r is a prime, NORM =2
> >>> -29 -21r is a prime or associated prime, norm =  -41
>
> >>> w+1 = r(-29 -21r )+1  = -41 -29r = unit
>
> >>> So given r, -29-21r, a finite set of primes from Euclid's
> >>> approach I cannot always conculde that there is an "extra" prime.
> >>> So in general, Euclids approach does not work.
>
> >>> How does what you say about fewunit rings apply to
> >>> "infinite" unit rings? If it doesn't, why is my claim wrong ?
>
> >> That ring has as many units as elts so its not a fewunit ring.
> >> The fewunits proof works for infinite rings with fewer units
> >> than elts, i.e. smaller cardinality.
>
> > Then why do you say "... In general  1 + pqr... can fail to
> > produce a new prime (or irreducible) because it may be a unit...".
>
> Because generally that's the obstruction to the Euclidean proof.

Yes, if you use nothing but the original construction in some other
infinite ring
and you can't exclude the possibility that if w+1 is a
unit then the proof does not go through.

> The fewunits hypothesis is one way of avoiding that obstruction.

If there wasn't an obstruction, you wouln;t need the generalization.
Euclid's proof would go through in its original form.

What is the obstruction you are referring to ?
..

> Specializing to Z (or any infinite UFD as in the theorem below)
> we see that the obstruction to this fewunits generalization of
> the Euclidean proof occurs only when there are infinitely many
> units -- not, as you claim, when there are units > 1. Even if
> Z had finitely many units>1 the fewunits proof would still work.

Presumably, you have to use the few units generalization
because the original Euclidean method does not work
as originally formulated. I never said that
the generalization of Euclid's proof wouldn't work.

>
>
> >>>> Such rings may have units besides +-1, so the w+1 candidates
> >>>> may indeed be units. But since there are fewer units than elts
> >>>> a pigeonhole argument shows that at least one candidate isn't
> >>>> a unit, so the proof does indeed succeed. Thus even for the
> >>>> ring of integers, the proof can succeed without employing
> >>>> the fact that there are no units>1. Instead one may employ
> >>>> the scarcity of units in order to avoid them as need be.
>
> >>>>> A unit has no prime divisors. If w+1 could be either a prime,
> >>>>> composite or unit, then its non-divisibility by all primes or
> >>>>> any prime leads to no contradiction or no "extra" prime
> >>>>> (w+1 is a unit) and the proof would not go through.
>
> >>>> But it does go through for infinite fewunit rings like Z.
> >>>> If the proof in [1] is beyond your knowledge of algebra
> >>>> (e.g. it uses simple ideal theory) you can find a much
> >>>> simpler element-wise version in [2]. That 4-line proof
> >>>> should be easily comprehensible to a high-school student.
> >>>> Indeed, here is a simple specialization of said proof:
>
> >>>> THEOREM  If Z is infinite UFD with finitely many units
> >>>> then Z has infinitely many primes.
>
> >>>> PROOF  Via  contradiction.  Let  w = product of all primes of Z
> >>>> Since  Z  is infinite so is  1 + w Z  [ 1+wn = 1+wn'  <=>  n = n']
> >>>> so it must contain a nonzero nonunit  [ since only finite #units ]
> >>>> which has a prime factor p | 1 + w n. But  p|w => p|1  =><=  QED
>
> >>> Yes, very good. How do you generalize Euclid's approach to rings
> >>> with infinitely many units ?
>
> >> See the penultimate paragraph in [2]
>
> --Bill Dubuque
>
> [1]http://www.mathlinks.ro/viewtopic.php?p=1209616http://google.com/groups?selm=y8zk5f3rn4e.fsf%40nestle.csail.mit.edu
>
> [2] sci.math, 12 Apr 2005, Euclid in rings: infinitely many primeshttp://google.com/group/sci.math/msg/c502d844bdaeee1ehttp://google.com/groups?selm=y8zbr8kv5vl.fsf_-_%40nestle.csail.mit.edu
>
> Beware: cut'n'paste error in above: instead of nilradical, please
> read pseudo-radical = intersection of all nonzero prime ideals.- Hide quoted text -
>
> - Show quoted text -- Hide quoted text -
>
> - Show quoted text -

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