I should amend my previous mail on this. Fist, why I wrote
about finite number of factorizations? The reason is that
when we have finite number of soultion to the equation system
coming from factorization, then one can find if solution is
in base field. In fact, simple method of filtering out
quantities involving algebraic extentions will give solutions
in base field. So provided we know that there is finite
number of factorizations one difficulty is gone.
Hopefully it is clear now why I care about finite number
of factorizations.
Konrad got upset that I wrote (about Cohn) "he jumps
over few subtle points". If I knew that anybody may
get upset about this I would not use those words.
Below is longer version that says the same thing in
more neutral language and gives concrete facts.
In "Free Ideal Rings and Localization in General Rings"
at start of section 4.3 (Conditions for a distributive
factor lattice) Cohn gives argument that I interpret
as proof of finite number of factorizations. In the
proof there are following sentences:
: Thus in a sense we have a one-parameter family of
: factorizations of c; this idea may be formalized by
: adjoining an indeterminate t to k and showing that
: (1) leads to a factorization of c in R \otimes k(t)
: that does not arise from a factorization in R (so
: that c is not inert in R \otimes k(t)).
To explain this a bit more, (1) refers to formula
c = ab, that is to fact that we have factorization
of c. The 'R \otimes k(t)' really means that we
extend base field by new transcendental parameter t.
My trouble is in the middle part of this fragment,
that is:
: showing that
: (1) leads to a factorization of c in R \otimes k(t)
: that does not arise from a factorization in R
I do not know how to show this and up to now I see
no clues in Cohn book how to prove it.
Of course, I would be glad if anybody could provide
me more elaborate proof (or reference to such proof).
--
Waldek Hebisch
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