malcobe@[EMAIL PROTECTED]
wrote:
>
> I keep on believing that the key idea [in Euclid's proof that there
> are infinitely many primes] is the construction of the number
> "the product of all primes plus one"
Actually the "reason" such proofs work is because the ring of integers
has relatively few units. Thus, by way of the Pigeonhole Principle,
one can avoid them and generate an infinite sequence of pairwise-coprime
nonunits. For the details see the generalizations given in the theorems
below from my prior post [0].
sttscitrans@[EMAIL PROTECTED]
wrote:
>
> Is there any analog A to N, where the usual rules of arithmetic
> apply, every non-unit is divisible by some prime, A is infinite
> and yet the number of primes or irreducibles are finite ?
Yes, vacuously, let A = Q or any infinite field. Less trivially
force all but finitely many primes to become units, e.g. grow
Z to a subring of Q by adjoining 1/p for almost all primes p.
For example, all primes but 2 become units in the subring of Q
of all rational numbers expressible in the form m/n with n odd.
All primes but 2,3 are units in subring of form m/n, (n,6) = 1.
Such enlarged domains remain Euclidean -> PID -> UFD... since
the localization construction always preserves such structure.
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, e.g.
THEOREM Suppose that D is an infinite integral domain where all
nonunits have an irreducible factor. If D has finitely many units
then D has infinitely many irreducibles (= primes if D is a UFD).
PROOF Via contradiction. Let m = product of all irreducibles.
Since D is infinite so is 1 + m D [ 1+md = 1+md' <=> d = d']
so it must contain a nonzero nonunit [ since only finite #units ]
with irreducible factor p | 1 + m d; but p|m => p|1 =><= QED
It is easy to generalize this theorem even further, for example:
THEOREM If a commutative ring R has a smaller cardinality subset U
containing all divisors of 0 and 1, then ~U, the complement of U,
contains an infinite number of pair-coprime nonunits (hence also an
infinite number of irreds if every r in ~U has an irred factor).
PROOF Enlarge any finite set P < ~U of pair-coprime nonunits as
follows. Let m be the product of all p in P (m = 1 if P empty).
Elements of S = 1 + m R are coprime to each p in P since p|m
Hence S is disjoint from P since coprime nonunits cannot be equal.
Now enlarge P by adjoining any s in S /\ ~U, which is nonempty
since S is too big to lie inside U by the cardinality hypothesis.
In fact #S = #R > #U because r -> 1 + m r injects R into S,
that is 1+mr = 1+mr' => mr = mr' => r = r' via m cancellable,
being a product of non-zero-divisor (so cancellable) p in P. QED
COROLLARY Elements of a finite commutative ring R divide 0 or 1.
In particular a finite integral domain is a field.
PROOF Let U = { all divisors of 0 and 1 }. Necessarily U = R
else #U < #R => ~U is infinite by Theorem, contra R finite. QED
What a surprisingly sweet little application of Euclid's theorem!
Compare the above proof with the classic pigeonhole based proof
http://planetmath.org/encyclopedia/AFiniteIntegralDomainIsAField.html
Other ring theoretic generalizations of Euclid's theorem can be
expressed in terms of the nilradical (intersection of all prime
ideals) and related notions such as G-domains [1]. If there are
only a finite number of prime ideals then the nilradical is
clearly nonzero: take a product of elements from each prime.
But for a wide class of domains, Krull domains (including UFDs,
Dedekind domains, etc) it is easy to show that the nilradical
is nonzero iff D has the structure in the example above, i.e.
a PID with finitely many primes. So, for example, any non-PID
in this class has infinitely many prime ideals; e.g. this
leads to Larry Wa****ngton's proof of Euclid's theorem via any
non-UFD number ring (if Z had a finite #primes then so would
every number ring, so every number ring would be a PID so UFD)
Closely related are various topological generalizations of
Euclid's theorem, e.g. work by Golomb, Gotchev, ****ubsky.
These are the sort of results that a curious student can easily
discover independently when pondering generalizations of Euclid's
theorem (indeed I found them as a teenager). They make excellent
exercises for beginning number theory students. If I can dig up
my original notes I will post further results. I hope readers
will contribute some of their favorites (there are many other
variants of proofs of Euclid's theorem listed in Ribenboim's
Book of Prime Number Records, but not those I presented above).
--Bill Dubuque
[0]
http://google.com/groups?selm=y8zbr8kv5vl.fsf_-_%40nestle.csail.mit.edu
[1] http://google.com/groups?selm=y8z7jjzdivj.fsf%40nestle.csail.mit.edu
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