Re: Dedekind Domain


Bill Dubuque wrote:
> Arturo Magidin <magidin@xxxxxxxxxxxxxxxxx> wrote:
> >Taedong Yun <noname@xxxxxxxxxxxxx> wrote:
> >>
> >> Let R be a domain with the property that every nonzero ideal
> >> is a product of maximal ideals. Show that R is Dedekind.
> >
> > This proof is in Jacobson's "Basic Algebra II", second edition,
> > Chapter 10, Section 2, Theorem 10.5. The proof is attributed to
> > Zariski and Samuel.
>
> This result is older than Zariski and Samuel's textbook (1958).

Sorry I wasn't clear. The ->proof<- in Jacobson's book is attributed to
Zariski and Samuel; and I mean the proof. The theorem is stated, and
the attribution is given after "Proof:", not in the statement of the
theorem. Since Jacobson defines Dedekind domain in terms of the
invertibiliity of ideals and via maximal ideals rather than in terms of
dimension/noetherian/etc or of unique factorization, it is possible
that this particular proof of this arrow in the argument establishing
that all sorts of definitions are equivalent is due to Zariski and
Samuel; it is probably that which Jacobson meant.

Arturo Magidin, sans .sig

.