domains with pairs of elements having no gcd
rabbits77
having no gcd?
I have started with this:
Let k be a field and let R be the subring of k[x] consisting of all
polynomials having no linear terms. That is , f(x) is in R iff
f(x)=s_0+s_2x^2+s_3x^3+.......
I think that if I show that x^5 and x^6 have no gcd in R I am done but
this is where I am stuck.
Terry E
"rabbits77" <rabb...@bigmailbox.net> wrote in message
news:68aadb4a.03050...@posting.google.com...
You might be in trouble, because I think that this may work. My batting
average lately has been poor, so when I say I think you are correct, you had
better let someone else check it over. That said,
Your ring R is a subring of k[x], imbed it into k[x]. Take your
candidates, x^5 and x^6 and find their gcd in k[x]. This does not have a
preimage. In other words, if x^5 and x^6 had a gcd in R, then this would be
a gcd in k[x] up to units in k[x]. Note that if I continue this and imbed
everything into k(x) then all non-zero elements are units, so I think this
is safe. If not, some kind soul will ask me about the medication I am on.
Terry.
Tralfaz
Q(sqrt[n]) is not a UFD, then one can find two elements with the
characteristic that for every common factor, there is another common factor
that does not divide it.
-Tralfaz
"rabbits77" <rabb...@bigmailbox.net> wrote in message
news:68aadb4a.03050...@posting.google.com...
Arturo Magidin
>Off the top of my head (so probably wrong), I seem to recall that if
>Q(sqrt[n]) is not a UFD, then one can find two elements with the
>characteristic that for every common factor, there is another common factor
>that does not divide it.
You are right... but that's because Q(sqrt(n)) is always a UFD. It's a
field.
You perhaps meant Z[sqrt(n)]?
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
mag...@math.berkeley.edu
Bill Dubuque
>Tralfaz <tra...@takeaguess.com> wrote:
>>>rabbits77 <rabb...@bigmailbox.net> wrote:
>>>
>>> Show there is a domain with a pair of elements having no gcd
>>
>> any Q(sqrt[n]) not a UFD [...]
>
> [but] Q(sqrt(n)) is always a UFD, being a field.
Presumably he meant this: for J = ring of integers of Q(sqrt(n))
J not UFD => J not GCD [gcd(x,y) fails to exist for some x,y]
More generally: for atomic domains: not UFD => not GCD domain
i.e. for atomic domains D is a UFD <=> D is a GCD domain
a well-known elementary characterization of UFDs (with proof
essentially the same as the standard proof that Z is a UFD).
Recall a domain D is atomic if every nonzero nonunit has a finite
factorization into atoms (irreducibles), and D is a GCD domain
if within D every nonzero pair x,y possess a gcd, denoted (x,y).
Number Rings are always atomic: the norm pulls it back from Z
(else an infinite factorization would be preserved by the norm).
There has been much study of domains related to GCD domains.
Below are some of them, in increasing order of generality.
PID: every ideal is principal
Bezout: every ideal (a,b) is principal
GCD: (x,y) := gcd(x,y) exists for all x,y
SCH: Schreier = pre-Schreier & integrally closed
SCH0: pre-Schreier: a|bc => a = BC, B|b, C|c
D: (a,b) = 1 & a|bc => a|c
PP: (a,b) = (a,c) = 1 => (a,bc) = 1
GL: Gauss Lemma: product of primitive polys is primitive
GL2: Gauss Lemma holds for all polys of degree 1
AP: atoms are prime [AP = PP restriced to atomic a]
PID => UFD These implications all hold true. In general,
| | no => reverses (counterexamples are known)
V V
Bezout => GCD => SCH => SCH0 => D => {PP <=> GL <=> GL2} => AP
Since atomic & AP => UFD, reversing the above UFD => AP path
shows that in atomic domains all these properties (except for
PID, Bezout) collapse, becoming all equivalent to being a UFD.
There are also many properties known equivalent to D, e.g.
[a] (a,b) = 1 => a|bc => a|c
[b] (a,b) = 1 => a,b|c => ab|c
[c] (a,b) = 1 => (a)/\(b) = (ab)
[d] (a,b) exists => lcm(a,b) exists
[e] a + bX irreducible => prime for b != 0 (deg = 1)
For much further discussion of these topics see the surveys [2],[3].
Both collect together many results which were previously scattered
widely throught the literature.
Recall that in an earlier post [1] I mentioned that Schreier domains
are precisely those domains for which holds true the main result of
the paper [5] of Magidin and Mckinnon. Reviewing that post reminds
me that I forgot to supply the reference to Paul Cohn's paper [4]
of 1968 which includes the main result of M&M. See also Anderson's
32 page survey [2], esp. pp. 7-8, 16-18. As I mentioned in [1],
I suspect the main M&M result dates back over 100 years ago to
Kronecker (or a contemporary) but, alas, I've had no luck jogging
my fuzzy memory. Hopefully Arturo will be able to cite Cohn's work
in his Monthly paper -- it deserves to be better known.
-Bill Dubuque
[1] http://groups.google.com/groups?&selm=y8zbs3larw2.fsf%40nestle.ai.mit.edu
[2] Anderson, D. D.(1-IA); Quintero, R. O.(YV-LAND2) MR 98i:13001
Some generalizations of GCD-domains.
Factorization in integral domains (Iowa City, IA, 1996), 189-195,
Lecture Notes in Pure and Appl. Math., 189, Dekker, New York, 1997.
[3] Anderson, D. D.(1-IA) MR 2002g:13039
GCD domains, Gauss' lemma, and contents of polynomials.
Non-Noetherian commutative ring theory, 1-31, Math. Appl., 520,
Kluwer Acad. Publ., Dordrecht, 2000.
[4] Cohn, P. M. MR 36#5117
Bezout rings and their subrings.
Proc. Cambridge Philos. Soc. 64 1968 251-264.
[5] Magidin, A.; Mckinnon, D.
Gauss's lemma for number fields
http://math.berkeley.edu/~magidin/preprints/gauss.ps
Arturo Magidin
Bill Dubuque <w...@nestle.ai.mit.edu> wrote:
[.snip.]
>There has been much study of domains related to GCD domains.
>Below are some of them, in increasing order of generality.
>
> PID: every ideal is principal
>
> Bezout: every ideal (a,b) is principal
>
> GCD: (x,y) := gcd(x,y) exists for all x,y
>
> SCH: Schreier = pre-Schreier & integrally closed
>
> SCH0: pre-Schreier: a|bc => a = BC, B|b, C|c
I asked George Bergman about this recently, and he told me that this
property is often refered to by saying that the lattice of ideals of R
has the "Riesz Interpolation Property" with respect to principal
ideals; at least, when studied from the point of view of properties of
the lattice of ideals.
[.snip.]
>Recall that in an earlier post [1] I mentioned that Schreier domains
>are precisely those domains for which holds true the main result of
>the paper [5] of Magidin and Mckinnon. Reviewing that post reminds
>me that I forgot to supply the reference to Paul Cohn's paper [4]
>of 1968 which includes the main result of M&M.
Ah! Thank you!
>Hopefully Arturo will be able to cite Cohn's work
>in his Monthly paper -- it deserves to be better known.
We have submitted the corrected manuscript, but I will try to get my
hand on a copy of Cohn's paper so that an explicit reference can be
given. Thank you very much!
maky m.
> How do I show that there are domains R containing a pair of elements
> having no gcd?
here is a rather cumbersome example i got from Joseph Rotman's _Galois
Theory_
let R be the ring of all functions f from IR into IR under pointwise
operations: if f, g are in IR, then
1. f+g (a) = f(a) + g(b),
2. f.g (a) = f(a) . g(b)
f and g defined by
f(x) = max{x, 0}
g(x) = min{x, 0}
have no gcd in R.
i tried asking the author why he chose such obscure example, but he
failed to provide an answer.